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

Theoremelequ2 1601 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

Theoremax11i 1602 Inference that has ax-11 1397 (without ) as its conclusion and doesn't require ax-10 1396, ax-11 1397, or ax-12 1402 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)

1.3.9  Axioms ax-10 and ax-11

Theoremax10o 1603 Show that ax-10o 1604 can be derived from ax-10 1396. An open problem is whether this theorem can be derived from ax-10 1396 and the others when ax-11 1397 is replaced with ax-11o 1704. See theorem ax10 1605 for the rederivation of ax-10 1396 from ax10o 1603.

Normally, ax10o 1603 should be used rather than ax-10o 1604, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)

Axiomax-10o 1604 Axiom ax-10o 1604 ("o" for "old") was the original version of ax-10 1396, before it was discovered (in May 2008) that the shorter ax-10 1396 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 1603.

Normally, ax10o 1603 should be used rather than ax-10o 1604, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremax10 1605 Rederivation of ax-10 1396 from original version ax-10o 1604. See theorem ax10o 1603 for the derivation of ax-10o 1604 from ax-10 1396.

This theorem should not be referenced in any proof. Instead, use ax-10 1396 above so that uses of ax-10 1396 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

Theoremhbae 1606 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theoremnfae 1607 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbaes 1608 Rule that applies hbae 1606 to antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremhbnae 1609 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.)

Theoremnfnae 1610 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbnaes 1611 Rule that applies hbnae 1609 to antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremnaecoms 1612 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)

Theoremequs4 1613 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)

Theoremequsalh 1614 A useful equivalence related to substitution. New proofs should use equsal 1615 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)

Theoremequsal 1615 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.)

Theoremequsex 1616 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theoremequsexd 1617 Deduction form of equsex 1616. (Contributed by Jim Kingdon, 29-Dec-2017.)

Theoremdral1 1618 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.)

Theoremdral2 1619 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.)

Theoremdrex2 1620 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.)

Theoremdrnf1 1621 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremdrnf2 1622 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremspimth 1623 Closed theorem form of spim 1626. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)

Theoremspimt 1624 Closed theorem form of spim 1626. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)

Theoremspimh 1625 Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1626 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.)

Theoremspim 1626 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1626 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.)

Theoremspimeh 1627 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.)

Theoremspimed 1628 Deduction version of spime 1629. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)

Theoremspime 1629 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.)

Theoremcbv3 1630 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)

Theoremcbv3h 1631 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.)

Theoremcbv1 1632 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.)

Theoremcbv1h 1633 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)

Theoremcbv2h 1634 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremcbv2 1635 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.)

Theoremcbvalh 1636 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremcbval 1637 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremcbvexh 1638 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)

Theoremcbvex 1639 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremchvar 1640 Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremequvini 1641 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.)

Theoremequveli 1642 A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1641.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)

Theoremnfald 1643 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.)

Theoremnfexd 1644 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.)

1.3.10  Substitution (without distinct variables)

Syntaxwsb 1645 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.)

Definitiondf-sb 1646 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 1658.

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 1721, sbcom2 1863 and sbid2v 1872).

Note that our definition is valid even when and are replaced with the same variable, as sbid 1657 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 1867 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 1870.

When and are distinct, we can express proper substitution with the simpler expressions of sb5 1767 and sb6 1766.

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.)

Theoremsbimi 1647 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)

Theoremsbbii 1648 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremsb1 1649 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsb2 1650 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ1 1651 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ2 1652 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremstdpc7 1653 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1591.) 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.)

Theoremsbequ12 1654 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ12r 1655 An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremsbequ12a 1656 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbid 1657 An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremstdpc4 1658 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.)

Theoremsbh 1659 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.)

Theoremsbf 1660 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.)

Theoremsbf2 1661 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)

Theoremsb6x 1662 Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremnfs1f 1663 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbs1f 1664 If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsbequ5 1665 Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.)

Theoremsbequ6 1666 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.)

Theoremsbt 1667 A substitution into a theorem remains true. (See chvar 1640 and chvarv 1812 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremequsb1 1668 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremequsb2 1669 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremsbiedh 1670 Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1673). New proofs should use sbied 1671 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)

Theoremsbied 1671 Conversion of implicit substitution to explicit substitution (deduction version of sbie 1674). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsbiedv 1672* Conversion of implicit substitution to explicit substitution (deduction version of sbie 1674). (Contributed by NM, 7-Jan-2017.)

Theoremsbieh 1673 Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1674 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)

Theoremsbie 1674 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.)

1.3.11  Theorems using axiom ax-11

Theoremequs5a 1675 A property related to substitution that unlike equs5 1710 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremequs5e 1676 A property related to substitution that unlike equs5 1710 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)

Theoremax11e 1677 Analogue to ax-11 1397 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.)

Theoremax10oe 1678 Quantifier Substitution for existential quantifiers. Analogue to ax10o 1603 but for rather than . (Contributed by Jim Kingdon, 21-Dec-2017.)

Theoremdrex1 1679 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.)

Theoremdrsb1 1680 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.)

Theoremexdistrfor 1681 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.)

Theoremsb4a 1682 A version of sb4 1713 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremequs45f 1683 Two ways of expressing substitution when is not free in . (Contributed by NM, 25-Apr-2008.)

Theoremsb6f 1684 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.)

Theoremsb5f 1685 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.)

Theoremsb4e 1686 One direction of a simplified definition of substitution that unlike sb4 1713 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb2a 1687 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb2e 1688 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb3 1689 If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.)

Theoremnfs1 1690 If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremsbcof2 1691 Version of sbco 1842 where is not free in . (Contributed by Jim Kingdon, 28-Dec-2017.)

1.4  Predicate calculus with distinct variables

1.4.1  Derive the axiom of distinct variables ax-16

Theoremspimv 1692* A version of spim 1626 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.)

Theoremaev 1693* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1695. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremax16 1694* Theorem showing that ax-16 1695 is redundant if ax-17 1419 is included in the axiom system. The important part of the proof is provided by aev 1693.

See ax16ALT 1739 for an alternate proof that does not require ax-10 1396 or ax-12 1402.

This theorem should not be referenced in any proof. Instead, use ax-16 1695 below so that theorems needing ax-16 1695 can be more easily identified. (Contributed by NM, 8-Nov-2006.)

Axiomax-16 1695* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1419 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 1419; see theorem ax16 1694.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1694. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremdveeq2 1696* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Theoremdveeq2or 1697* Quantifier introduction when one pair of variables is distinct. Like dveeq2 1696 but connecting by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.)

TheoremdvelimfALT2 1698* Proof of dvelimf 1891 using dveeq2 1696 (shown as the last hypothesis) instead of ax-12 1402. This shows that ax-12 1402 could be replaced by dveeq2 1696 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.)

Theoremnd5 1699* A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.)

Theoremexlimdv 1700* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)

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