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Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem19.35ri 1601 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.25 1602 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.30 1603 Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem19.43 1604 Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)

Theorem19.43OLD 1605 Obsolete proof of 19.43 1604 as of 3-May-2016. Leave this in for the example on the mmrecent.html page. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem19.33 1606 Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.33b 1607 The antecedent provides a condition implying the converse of 19.33 1606. Compare Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.)

Theorem19.40 1608 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.40-2 1609 Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theoremalrot3 1610 Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)

Theoremalrot4 1611 Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)

Theoremalbiim 1612 Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)

Theorem2albiim 1613 Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.)

Theoremhbald 1614 Deduction form of bound-variable hypothesis builder hbal 1567. (Contributed by NM, 2-Jan-2002.)

Theoremexintrbi 1615 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)

Theoremexintr 1616 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)

Theoremalsyl 1617 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)

1.5.2  Introduce equality axioms ax-8, ax-11, ax-13, and ax-14

Syntaxcv 1618 This syntax construction states that a variable , which has been declared to be a set variable by \$f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder is a class by cab 2239. Since (when is distinct from ) we have by cvjust 2248, we can argue that that the syntax " " can be viewed as an abbreviation for " ". See the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class."

While it is tempting and perhaps occasionally useful to view cv 1618 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1618 is intrinsically no different from any other class-building syntax such as cab 2239, cun 3076, or c0 3362.

For a general discussion of the theory of classes and the role of cv 1618, see http://us.metamath.org/mpegif/mmset.html#class.

(The description above applies to set theory, not predicate calculus. The purpose of introducing here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1620 of predicate calculus from the wceq 1619 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.)

Syntaxwceq 1619 Extend wff definition to include class equality.

For a general discussion of the theory of classes, see http://us.metamath.org/mpegif/mmset.html#class.

(The purpose of introducing here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1620 of predicate calculus in terms of the wceq 1619 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the in could be the of either weq 1620 or wceq 1619, although mathematically it makes no difference. The class variables and are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2246 for more information on the set theory usage of wceq 1619.)

Theoremweq 1620 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 1620 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wceq 1619. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1620 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1619. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

Syntaxwcel 1621 Extend wff definition to include the membership connective between classes.

For a general discussion of the theory of classes, see http://us.metamath.org/mpegif/mmset.html#class.

(The purpose of introducing here is to allow us to express i.e. "prove" the wel 1622 of predicate calculus in terms of the wceq 1619 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables and are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2240 for more information on the set theory usage of wcel 1621.)

Theoremwel 1622 Extend wff definition to include atomic formulas with the epsilon (membership) predicate. This is read " is an element of ," " is a member of ," " belongs to ," or " contains ." Note: The phrase " includes " means " is a subset of ;" to use it also for , as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT).

This syntactical construction introduces a binary non-logical predicate symbol (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments.

(Instead of introducing wel 1622 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 1621. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1622 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1621. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

Axiomax-8 1623 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. This is similar to, but not quite, a transitive law for equality (proved later as equtr 1826). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105.

This is our first use of the equality symbol, invented in 1527 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle."

Axioms ax-8 1623 through ax-16 1926 are the axioms having to do with equality, substitution, and logical properties of our binary predicate (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1926 and ax-17 1628 are still valid even when , , and are replaced with the same variable because they do not have any distinct variable (Metamath's \$d) restrictions. Distinct variable restrictions are required for ax-16 1926 and ax-17 1628 only. (Contributed by NM, 5-Aug-1993.)

Axiomax-11 1624 Axiom of Variable Substitution. One of the 5 equality axioms of predicate calculus. The final consequent is a way of expressing " substituted for in wff " (cf. sb6 1992). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases.

The original version of this axiom was ax-11o 1940 ("o" for "old") and was replaced with this shorter ax-11 1624 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 1939. Conversely, this axiom is proved from ax-11o 1940 as theorem ax11 1941.

Juha Arpiainen proved the independence of this axiom (in the form of the older axiom ax-11o 1940) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

Interestingly, if the wff expression substituted for contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of axiom ax-11o 1940 (from which the ax-11 1624 instance follows by theorem ax11 1941.) The proof is by induction on formula length, using ax11eq 2105 and ax11el 2106 for the basis steps and ax11indn 2108, ax11indi 2109, and ax11inda 2113 for the induction steps. (This paragraph is true provided we use ax-10o 1835 in place of ax-10 1678.)

See also ax11v 1990 and ax11v2 1935 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions. (Contributed by NM, 22-Jan-2007.)

Axiomax-13 1625 Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of the binary predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.)

Axiomax-14 1626 Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.)

Theoremequs3 1627 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

1.5.3  Axiom ax-17 - first use of the \$d distinct variable statement

Axiomax-17 1628* Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

This axiom is logically redundant in the (logically complete) predicate calculus axiom system consisting of ax-gen 1536, ax-4 1692 through ax-9 1684, ax-10o 1835, and ax-12o 1664 through ax-16 1926: in that system, we can derive any instance of ax-17 1628 not containing wff variables by induction on formula length, using ax17eq 1927 and ax17el 2103 for the basis together hbn 1722, hbal 1567, and hbim 1723. However, if we omit this axiom, our development would be quite inconvenient since we could work only with specific instances of wffs containing no wff variables - this axiom introduces the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct). (Contributed by NM, 5-Aug-1993.)

Theoremnfv 1629* If is not present in , then is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theorema17d 1630* ax-17 1628 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)

Theoremnfvd 1631* nfv 1629 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1727. (Contributed by Mario Carneiro, 6-Oct-2016.)

1.5.4  Introduce equality axioms ax-9v and ax-12

Axiomax-9v 1632* Axiom B8 of [Tarski] p. 75, which is the same as our axiom ax-9 1684 weakened with a distinct variable condition. Theorem ax9 1683 shows the derivation of ax-9 1684 from this one. (Contributed by NM, 7-Nov-2015.) (New usage is discouraged.)

Axiomax-12 1633 Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality.

The antecedents imply as well as , so must be a different variable from both and if the antecedents hold. The axiom essentially states that if is neither nor , then it is irrelevant to the truth of .

The original version of this axiom was ax-12o 1664 ("o" for "old") and was replaced with this shorter ax-12 1633 in December 2015. The old axiom is proved from this one as theorem ax12o 1663. Conversely, this axiom is proved from ax-12o 1664 as theorem ax12 1881.

Although this version is shorter, the original version ax-12o 1664 may be more intuitive to understand, as well as more practical to work with, because of the "distinctor" form of its antecedents.

This axiom can be weakened if desired by adding distinct variable restrictions on pairs and . To show that, we add these restrictions to theorem ax12v 1634 use only ax12v 1634 for further derivation. Thus ax12v 1634 should be the only theorem referencing this axiom. Other theorems can reference either ax12v 1634 or ax12 1881. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)

Theoremax12v 1634* A weaker version of ax-12 1633 with distinct variable restrictions on pairs and . In order to show that this weakening is adequate, this should be the only theorem referencing ax-12 1633 directly. (Contributed by NM, 30-Jun-2016.)

1.5.5  Derive ax-12o from ax-12

Theoremax12o10lem1 1635 Lemma for ax12o 1663 and ax10 1677. Same as equcomi 1822, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632.

Note that in these lemmas we use ax-9v 1632 instead of ax-9 1684 since the proof of ax9 1683 from ax-9v 1632 makes use of ax-12o 1664. The first use of ax-12o 1664 occurs in ax10lem24 1673. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax12o10lem2 1636 Lemma for ax12o 1663 and ax10 1677. Same as equequ1 1829, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax12o10lem3 1637 Lemma for ax12o 1663 and ax10 1677. Same as ax4 1691, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax12o10lem4 1638 Lemma for ax12o 1663. Same as 19.8a 1758, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 7-Nov-2015.) (New usage is discouraged.)

Theoremax12o10lem5 1639 Lemma for ax12o 1663 and ax10 1677. Same as hba1 1718, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax12o10lem6 1640 Lemma for ax12o 1663 and ax10 1677. Same as hbn 1722, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax12o10lem7 1641 Lemma for ax12o 1663 and ax10 1677. Same as hbimd 1809, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax12o10lem8 1642* Lemma for ax12o 1663 and ax10 1677. Similar to a4ime 1868 with distinct variables, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax12o10lem9 1643 Lemma for ax12o 1663. Same as ax6o 1696 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)

Theoremax12o10lem10 1644 Lemma for ax12o 1663. Same as hbnt 1717 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)

Theoremax12o10lem11 1645 Lemma for ax12o 1663. Same as hbim 1723 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)

Theoremax12o10lem12 1646 Lemma for ax12o 1663. Same as 19.9t 1761 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)

Theoremax12o10lem13 1647 Lemma for ax12o 1663. Same as 19.9 1762 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)

Theoremax12o10lem14 1648 Lemma for ax12o 1663. Same as 19.23 1777 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)

Theoremax12o10lem15 1649 Lemma for ax12o 1663. Same as exlimih 1782 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)

Theoremax12olem16 1650* Lemma for ax12o 1663. Weaker version of equsalh 1851 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)

Theoremax12olem17 1651 Lemma for ax12o 1663. Same as 19.21 1771 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)

Theoremax12olem18 1652 Lemma for ax12o 1663. Same as hbim1 1810 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)

Theoremax12olem19 1653 Lemma for ax12o 1663. Same as nfex 1733 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 20-Dec-2015.) (New usage is discouraged.)

Theoremax12olem20 1654 Lemma for ax12o 1663. Same as 19.12 1766 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 20-Dec-2015.) (New usage is discouraged.)

Theoremax12olem21 1655* Lemma for ax12o 1663. Similar to equvin 1999 but with a negated equality. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.)

Theoremax12olem22 1656* Lemma for ax12o 1663. Negate the equalities in ax-12 1633, shown as the hypothesis. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.)

Theoremax12olem23 1657 Lemma for ax12o 1663. Show the equivalence of an intermediate equivalent to ax-12o 1664 with the conjunction of ax-12 1633 and a variant with negated equalities. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.)

Theoremax12olem24 1658* Lemma for ax12o 1663. Construct an intermediate equivalent to ax-12 1633 from two instances of ax-12 1633. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.)

Theoremax12olem25 1659 Lemma for ax12o 1663. See ax12olem27 1661 for derivation of ax-12o 1664 from the conclusion. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.)

Theoremax12olem26 1660* Lemma for ax12o 1663. Same as dvelimfALT2 27814 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.) (New usage is discouraged.)

Theoremax12olem27 1661* Lemma for ax12o 1663. Derivation of ax-12o 1664 from the hypotheses, without using ax-12o 1664. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.) (New usage is discouraged.)

Theoremax12olem28 1662* Lemma for ax12o 1663. Derivation of ax-12o 1664 from the hypotheses, without using ax-12o 1664. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.)

Theoremax12o 1663 Derive set.mm's original ax-12o 1664 from the shorter ax-12 1633.

Our current practice is to use axiom ax-12o 1664 from here on (except in the proofs of ax10 1677 and ax9 1683 below) instead of theorem ax12o 1663 in order to standardize the use ax-9 1684 instead of ax-9v 1632. Note that the derivation of ax-9 1684 from ax-9v 1632 (theorem ax9 1683 below) makes use of ax12o 1663; thus we use ax-9v 1632 to prove ax12o 1663 to avoid a circular argument . (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (New usage is discouraged.)

Axiomax-12o 1664 Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever is distinct from and , and is true, then quantified with is also true. In other words, is irrelevant to the truth of . Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

The analogous axiom for the membership connective, ax-15 2102, has been shown to be redundant (theorem ax15 2101).

In December 2015, this axiom was replaced with a shorter version, ax-12 1633. Theorem ax12o 1663 shows the derivation of ax-12o 1664 from ax-12 1633, and theorem ax12 1881 shows the reverse derivation. (Contributed by NM, 5-Aug-1993.)

1.5.6  Derive ax-10

Theoremax10lem16 1665 Lemma for ax10 1677. Similar to equequ2 1830, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax10lem17 1666 Lemma for ax10 1677. Similar to hban 1724, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax10lem18 1667 Lemma for ax10 1677. Similar to exlimih 1782, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax10lem19 1668* Lemma for ax10 1677. Similar to cbv3ALT 1875 with distinct variables, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax10lem20 1669* Change bound variable without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 22-Jul-2015.) (New usage is discouraged.)

Theoremax10lem21 1670* Change free variable without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax10lem22 1671* Lemma for ax10 1677. Similar to ax-10 1678 but with distinct variables, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax10lem23 1672 Lemma for ax10 1677. Similar to ax-10o 1835 but with reversed antecedent, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax10lem24 1673* Lemma for ax10 1677. Similar to dvelim 2092 with first hypothesis replaced by distinct variable condition, without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremax10lem25 1674* Lemma for ax10 1677. Similar to dveeq2 1928, without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (Contributed by NM, 20-Jul-2015.) (New usage is discouraged.)

Theoremax10lem26 1675* Distinctor with bound variable change without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (Contributed by NM, 8-Jul-2016.) (New usage is discouraged.)

Theoremax10lem27 1676* Change free and bound variables without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (Contributed by NM, 22-Jul-2015.) (New usage is discouraged.)

Theoremax10 1677 Proof of axiom ax-10 1678 from others, without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (See remarks for ax12o10lem1 1635 about why we use ax-9v 1632 instead of ax-9 1684.)

Our current practice is to use axiom ax-10 1678 from here on instead of theorem ax10 1677, in order to preferentially use ax-9 1684 instead of ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.) (New usage is discouraged.)

Axiomax-10 1678 Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).

The original version of this axiom was ax-10o 1835 ("o" for "old") and was replaced with this shorter ax-10 1678 in May 2008. The old axiom is proved from this one as theorem ax10o 1834. Conversely, this axiom is proved from ax-10o 1835 as theorem ax10from10o 1836.

This axiom was proved redundant in July 2015. See theorem ax10 1677. (Contributed by NM, 16-May-2008.)

Theorema16gALT 1679* Alternate proof of a16g 2000 without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

Theoremalequcom 1680 Commutation law for identical variable specifiers. The antecedent and consequent are true when and are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremalequcoms 1681 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)

Theoremnalequcoms 1682 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)

1.5.7  Derive ax-9 from the weaker version ax-9v

Theoremax9 1683 Theorem showing that ax-9 1684 follows from the weaker version ax-9v 1632.

This theorem normally should not be referenced in any later proof. Instead, the use of ax-9 1684 below is preferred, since it is easier to work with (it has no distinct variable conditions) and it is the standard version we have adopted. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (New usage is discouraged.)

1.5.8  Introduce Axiom of Existence ax-9

Axiomax-9 1684 Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. One thing this axiom tells us is that at least one thing exists (although ax-4 1692 and possibly others also tell us that, i.e. they are not valid in the empty domain of a "free logic"). In this form (not requiring that and be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) It is equivalent to axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is established by ax9o 1814 and ax9from9o 1816. A more convenient form of this axiom is a9e 1817, which has additional remarks.

Raph Levien proved the independence of this axiom from the other logical axioms on 12-Apr-2005. See item 16 at http://us.metamath.org/award2003.html.

ax-9 1684 can be proved from a weaker version requiring that the variables be distinct; see theorem ax9 1683.

ax-9 1684 can also be proved from the Axiom of Separation (in the form that we use that axiom, where free variables are not universally quantified). See theorem ax9vsep 4042. (Contributed by NM, 5-Aug-1993.)

Theoremax9v 1685* Rederivation of ax-9v 1632 from ax-9 1684. This trivial proof is intended merely to show how we can do this simply by adding a (redundant) distinct variable restriction. Together with theorem ax9 1683, it thus shows the equivalence (in the presence of the other axioms) of ax-9 1684 and ax-9v 1632. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.)

Theoremequidqe 1686 equid 1818 with existential quantifier without using ax-4 1692 or ax-17 1628. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.)

Theoremhbequid 1687 Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (The proof uses only ax-5 1533, ax-8 1623, ax-12o 1664, and ax-gen 1536. This shows that this can be proved without ax-9 1684, even though the theorem equid 1818 cannot be. A shorter proof using ax-9 1684 is obtainable from equid 1818 and hbth 1557.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax-9v 1632, which is used for the derivation of ax12o 1663, unless we consider ax-12o 1664 the starting axiom rather than ax-12 1633. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.)

Theoremnfequid 1688 Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (The proof uses only ax-5 1533, ax-8 1623, ax-12o 1664, and ax-gen 1536. This shows that this can be proved without ax-9 1684, even though the theorem equid 1818 cannot be. A shorter proof using ax-9 1684 is obtainable from equid 1818 and hbth 1557.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax-9v 1632, which is used for the derivation of ax12o 1663, unless we consider ax-12o 1664 the starting axiom rather than ax-12 1633. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.)

Theoremequidq 1689 equid 1818 with universal quantifier without using ax-4 1692 or ax-17 1628. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.)

Theoremax4sp1 1690 A special case of ax-4 1692 without using ax-4 1692 or ax-17 1628. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.)

1.5.9  Derive ax-4, ax-5o, and ax-6o

Theoremax4 1691 Theorem showing that ax-4 1692 can be derived from ax-5 1533, ax-gen 1536, ax-8 1623, ax-9 1684, ax-11 1624, and ax-17 1628 and is therefore redundant in a system including these axioms. The proof uses ideas from the proof of Lemma 21 of [Monk2] p. 114.

This theorem should not be referenced in any proof. Instead, we will use ax-4 1692 below so that explicit uses of ax-4 1692 can be more easily identified. In particular, this will more cleanly separate out the theorems of "pure" predicate calculus that don't involve equality or distinct variables. A beginner may wish to accept ax-4 1692 a priori, so that the proof of this theorem (ax4 1691), which involves equality as well as the distinct variable requirements of ax-17 1628, can be put off until those axioms are studied.

Note: All predicate calculus axioms introduced from this point forward are redundant. Immediately before their introduction, we prove them from earlier axioms to demonstrate their redundancy. Specifically, redundant axioms ax-4 1692, ax-5o 1694, ax-6o 1697, ax-9o 1815, ax-10o 1835, ax-11o 1940, ax-15 2102, and ax-16 1926 are proved by theorems ax4 1691, ax5o 1693, ax6o 1696, ax9o 1814, ax10o 1834, ax11o 1939, ax15 2101, and ax16 1925. Except for the ones suffixed with o (ax-5o 1694 etc.), we never reference those theorems directly. Instead, we use the axiom version that immediately follows it. This allow us to better isolate the uses of the redundant axioms for easier study of subsystems containing them.

Axioms ax-12o 1664, ax-10 1678, and ax-9 1684 should be referenced directly rather than theorems ax12o 1663, ax10 1677, and ax9 1683 that prove them from others. This will avoid the general use of ax-9v 1632, which is not an official axiom for us. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Axiomax-4 1692 Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all , it is true for any specific (that would typically occur as a free variable in the wff substituted for ). (A free variable is one that does not occur in the scope of a quantifier: and are both free in , but only is free in .) This is one of the axioms of what we call "pure" predicate calculus (ax-4 1692 through ax-7 1535 plus rule ax-gen 1536). Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).

Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1536. Conditional forms of the converse are given by ax-12 1633, ax-15 2102, ax-16 1926, and ax-17 1628.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1896.

An interesting alternate axiomatization uses ax467 1752 and ax-5o 1694 in place of ax-4 1692, ax-5 1533, ax-6 1534, and ax-7 1535.

This axiom is redundant in the presence of certain other axioms, as shown by theorem ax4 1691. (We replaced the older ax-5o 1694 and ax-6o 1697 with newer versions ax-5 1533 and ax-6 1534 in order to prove this redundancy.) (Contributed by NM, 5-Aug-1993.)

Theoremax5o 1693 Show that the original axiom ax-5o 1694 can be derived from ax-5 1533 and others. See ax5 1695 for the rederivation of ax-5 1533 from ax-5o 1694.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114.

Normally, ax5o 1693 should be used rather than ax-5o 1694, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)

Axiomax-5o 1694 Axiom of Quantified Implication. This axiom moves a quantifier from outside to inside an implication, quantifying . Notice that must not be a free variable in the antecedent of the quantified implication, and we express this by binding to "protect" the axiom from a containing a free . One of the 4 axioms of "pure" predicate calculus. Axiom scheme C4' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Lemma 5 of [Monk2] p. 108 and Axiom 5 of [Mendelson] p. 69.

This axiom is redundant, as shown by theorem ax5o 1693.

Normally, ax5o 1693 should be used rather than ax-5o 1694, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremax5 1695 Rederivation of axiom ax-5 1533 from the orginal version, ax-5o 1694. See ax5o 1693 for the derivation of ax-5o 1694 from ax-5 1533.

This theorem should not be referenced in any proof. Instead, use ax-5 1533 above so that uses of ax-5 1533 can be more easily identified.

Note: This is the same as theorem alim 1548 below. It is proved separately here so that it won't be dependent on the axioms used for alim 1548. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax6o 1696 Show that the original axiom ax-6o 1697 can be derived from ax-6 1534 and others. See ax6 1698 for the rederivation of ax-6 1534 from ax-6o 1697.

Normally, ax6o 1696 should be used rather than ax-6o 1697, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

Axiomax-6o 1697 Axiom of Quantified Negation. This axiom is used to manipulate negated quantifiers. One of the 4 axioms of pure predicate calculus. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). An alternate axiomatization could use ax467 1752 in place of ax-4 1692, ax-6o 1697, and ax-7 1535.

This axiom is redundant, as shown by theorem ax6o 1696.

Normally, ax6o 1696 should be used rather than ax-6o 1697, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremax6 1698 Rederivation of axiom ax-6 1534 from the orginal version, ax-6o 1697. See ax6o 1696 for the derivation of ax-6o 1697 from ax-6 1534.

This theorem should not be referenced in any proof. Instead, use ax-6 1534 above so that uses of ax-6 1534 can be more easily identified. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

1.5.10  "Pure" predicate calculus including ax-4, without distinct variables

Theorema4i 1699 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)

Theorema4s 1700 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)

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