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Type | Label | Description |
---|---|---|
Statement | ||
Axiom | ax-10o 1601 |
Axiom ax-10o 1601 ("o" for "old") was the
original version of ax-10 1393,
before it was discovered (in May 2008) that the shorter ax-10 1393 could
replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of
the preprint).
This axiom is redundant, as shown by theorem ax10o 1600. Normally, ax10o 1600 should be used rather than ax-10o 1601, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | ax10 1602 |
Rederivation of ax-10 1393 from original version ax-10o 1601. See theorem
ax10o 1600 for the derivation of ax-10o 1601 from ax-10 1393.
This theorem should not be referenced in any proof. Instead, use ax-10 1393 above so that uses of ax-10 1393 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.) |
Theorem | hbae 1603 | All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | nfae 1604 | All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbaes 1605 | Rule that applies hbae 1603 to antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | hbnae 1606 | All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.) |
Theorem | nfnae 1607 | All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbnaes 1608 | Rule that applies hbnae 1606 to antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | naecoms 1609 | A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) |
Theorem | equs4 1610 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) |
Theorem | equsalh 1611 | A useful equivalence related to substitution. New proofs should use equsal 1612 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
Theorem | equsal 1612 | A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.) |
Theorem | equsex 1613 | A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | equsexd 1614 | Deduction form of equsex 1613. (Contributed by Jim Kingdon, 29-Dec-2017.) |
Theorem | dral1 1615 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.) |
Theorem | dral2 1616 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) |
Theorem | drex2 1617 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) |
Theorem | drnf1 1618 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
Theorem | drnf2 1619 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
Theorem | spimth 1620 | Closed theorem form of spim 1623. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.) |
Theorem | spimt 1621 | Closed theorem form of spim 1623. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.) |
Theorem | spimh 1622 | Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1623 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 8-May-2008.) (New usage is discouraged.) |
Theorem | spim 1623 | Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1623 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.) |
Theorem | spimeh 1624 | Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.) |
Theorem | spimed 1625 | Deduction version of spime 1626. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) |
Theorem | spime 1626 | Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) |
Theorem | cbv3 1627 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) |
Theorem | cbv3h 1628 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.) |
Theorem | cbv1 1629 | Rule used to change bound variables, using implicit substitution. Revised to format hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) |
Theorem | cbv1h 1630 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.) |
Theorem | cbv2h 1631 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | cbv2 1632 | Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) |
Theorem | cbvalh 1633 | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | cbval 1634 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) |
Theorem | cbvexh 1635 | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.) |
Theorem | cbvex 1636 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | chvar 1637 | Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) |
Theorem | equvini 1638 | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require to be distinct from and (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | equveli 1639 | A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1638.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.) |
Theorem | nfald 1640 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) |
Theorem | nfexd 1641 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 7-Feb-2018.) |
Syntax | wsb 1642 | Extend wff definition to include proper substitution (read "the wff that results when is properly substituted for in wff "). (Contributed by NM, 24-Jan-2006.) |
Definition | df-sb 1643 |
Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the
preprint). For our notation, we use to mean "the wff
that results when
is properly substituted for in the wff
." We
can also use in
place of the "free for"
side condition used in traditional predicate calculus; see, for example,
stdpc4 1655.
Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, " is the wff that results when is properly substituted for in ." For example, if the original is , then is , from which we obtain that is . So what exactly does mean? Curry's notation solves this problem. In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 1718, sbcom2 1860 and sbid2v 1869). Note that our definition is valid even when and are replaced with the same variable, as sbid 1654 shows. We achieve this by having free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 1864 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 1867. When and are distinct, we can express proper substitution with the simpler expressions of sb5 1764 and sb6 1763. In classical logic, another possible definition is but we do not have an intuitionistic proof that this is equivalent. There are no restrictions on any of the variables, including what variables may occur in wff . (Contributed by NM, 5-Aug-1993.) |
Theorem | sbimi 1644 | Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) |
Theorem | sbbii 1645 | Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb1 1646 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb2 1647 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ1 1648 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ2 1649 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | stdpc7 1650 | One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1588.) Translated to traditional notation, it can be read: " , , , provided that is free for in , ." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.) |
Theorem | sbequ12 1651 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ12r 1652 | An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Theorem | sbequ12a 1653 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbid 1654 | An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.) |
Theorem | stdpc4 1655 | The specialization axiom of standard predicate calculus. It states that if a statement holds for all , then it also holds for the specific case of (properly) substituted for . Translated to traditional notation, it can be read: " , provided that is free for in ." Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbh 1656 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.) |
Theorem | sbf 1657 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sbf2 1658 | Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.) |
Theorem | sb6x 1659 | Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | nfs1f 1660 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbs1f 1661 | If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sbequ5 1662 | Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.) |
Theorem | sbequ6 1663 | Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.) |
Theorem | sbt 1664 | A substitution into a theorem remains true. (See chvar 1637 and chvarv 1809 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | equsb1 1665 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | equsb2 1666 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbiedh 1667 | Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1670). New proofs should use sbied 1668 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
Theorem | sbied 1668 | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1671). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sbiedv 1669* | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1671). (Contributed by NM, 7-Jan-2017.) |
Theorem | sbieh 1670 | Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1671 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.) |
Theorem | sbie 1671 | Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Revised by Wolf Lammen, 30-Apr-2018.) |
Theorem | equs5a 1672 | A property related to substitution that unlike equs5 1707 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | equs5e 1673 | A property related to substitution that unlike equs5 1707 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.) |
Theorem | ax11e 1674 | Analogue to ax-11 1394 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.) |
Theorem | ax10oe 1675 | Quantifier Substitution for existential quantifiers. Analogue to ax10o 1600 but for rather than . (Contributed by Jim Kingdon, 21-Dec-2017.) |
Theorem | drex1 1676 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) (Revised by NM, 3-Feb-2015.) |
Theorem | drsb1 1677 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 5-Aug-1993.) |
Theorem | exdistrfor 1678 | Distribution of existential quantifiers, with a bound-variable hypothesis saying that is not free in , but can be free in (and there is no distinct variable condition on and ). (Contributed by Jim Kingdon, 25-Feb-2018.) |
Theorem | sb4a 1679 | A version of sb4 1710 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | equs45f 1680 | Two ways of expressing substitution when is not free in . (Contributed by NM, 25-Apr-2008.) |
Theorem | sb6f 1681 | Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.) |
Theorem | sb5f 1682 | Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.) |
Theorem | sb4e 1683 | One direction of a simplified definition of substitution that unlike sb4 1710 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb2a 1684 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb2e 1685 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb3 1686 | If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | nfs1 1687 | If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | sbcof2 1688 | Version of sbco 1839 where is not free in . (Contributed by Jim Kingdon, 28-Dec-2017.) |
Theorem | spimv 1689* | A version of spim 1623 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.) |
Theorem | aev 1690* | A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1692. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Theorem | ax16 1691* |
Theorem showing that ax-16 1692 is redundant if ax-17 1416 is included in the
axiom system. The important part of the proof is provided by aev 1690.
See ax16ALT 1736 for an alternate proof that does not require ax-10 1393 or ax-12 1399. This theorem should not be referenced in any proof. Instead, use ax-16 1692 below so that theorems needing ax-16 1692 can be more easily identified. (Contributed by NM, 8-Nov-2006.) |
Axiom | ax-16 1692* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1416 to be a
metatheorem and not an axiom). Axiom scheme C16' 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. It is a somewhat bizarre axiom since the antecedent is always
false in set theory, but nonetheless it is technically necessary as you
can see from its uses.
This axiom is redundant if we include ax-17 1416; see theorem ax16 1691. This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1691. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | dveeq2 1693* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | dveeq2or 1694* | Quantifier introduction when one pair of variables is distinct. Like dveeq2 1693 but connecting by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.) |
Theorem | dvelimfALT2 1695* | Proof of dvelimf 1888 using dveeq2 1693 (shown as the last hypothesis) instead of ax-12 1399. This shows that ax-12 1399 could be replaced by dveeq2 1693 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) |
Theorem | nd5 1696* | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.) |
Theorem | exlimdv 1697* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
Theorem | ax11v2 1698* | Recovery of ax11o 1700 from ax11v 1705 without using ax-11 1394. The hypothesis is even weaker than ax11v 1705, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1700. (Contributed by NM, 2-Feb-2007.) |
Theorem | ax11a2 1699* | Derive ax-11o 1701 from a hypothesis in the form of ax-11 1394. The hypothesis is even weaker than ax-11 1394, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1700. (Contributed by NM, 2-Feb-2007.) |
Theorem | ax11o 1700 |
Derivation of set.mm's original ax-11o 1701 from the shorter ax-11 1394 that
has replaced it.
An open problem is whether this theorem can be proved without relying on ax-16 1692 or ax-17 1416. Normally, ax11o 1700 should be used rather than ax-11o 1701, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.) |
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