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

Theoremeeor 1601 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)

1.3.8  Equality theorems without distinct variables

Theorema9e 1602 At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1352 through ax-14 1421 and ax-17 1435, all axioms other than ax-9 1440 are believed to be theorems of free logic, although the system without ax-9 1440 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theorema9ev 1603* At least one individual exists. Weaker version of a9e 1602. (Contributed by NM, 3-Aug-2017.)

Theoremax9o 1604 An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theoremequid 1605 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms.

This proof is similar to Tarski's and makes use of a dummy variable . It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.)

Theoremnfequid 1606 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. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)

Theoremstdpc6 1607 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1669.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)

Theoremequcomi 1608 Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)

Theoremequcom 1609 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)

Theoremequcoms 1610 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)

Theoremequtr 1611 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)

Theoremequtrr 1612 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)

Theoremequtr2 1613 A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremequequ1 1614 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)

Theoremequequ2 1615 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)

Theoremelequ1 1616 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

Theoremelequ2 1617 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

Theoremax11i 1618 Inference that has ax-11 1413 (without ) as its conclusion and doesn't require ax-10 1412, ax-11 1413, or ax-12 1418 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 1619 Show that ax-10o 1620 can be derived from ax-10 1412. An open problem is whether this theorem can be derived from ax-10 1412 and the others when ax-11 1413 is replaced with ax-11o 1720. See theorem ax10 1621 for the rederivation of ax-10 1412 from ax10o 1619.

Normally, ax10o 1619 should be used rather than ax-10o 1620, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)

Axiomax-10o 1620 Axiom ax-10o 1620 ("o" for "old") was the original version of ax-10 1412, before it was discovered (in May 2008) that the shorter ax-10 1412 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 1619.

Normally, ax10o 1619 should be used rather than ax-10o 1620, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremax10 1621 Rederivation of ax-10 1412 from original version ax-10o 1620. See theorem ax10o 1619 for the derivation of ax-10o 1620 from ax-10 1412.

This theorem should not be referenced in any proof. Instead, use ax-10 1412 above so that uses of ax-10 1412 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

Theoremhbae 1622 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theoremnfae 1623 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbaes 1624 Rule that applies hbae 1622 to antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremhbnae 1625 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 1626 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbnaes 1627 Rule that applies hbnae 1625 to antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremnaecoms 1628 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)

Theoremequs4 1629 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)

Theoremequsalh 1630 A useful equivalence related to substitution. New proofs should use equsal 1631 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)

Theoremequsal 1631 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 1632 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theoremequsexd 1633 Deduction form of equsex 1632. (Contributed by Jim Kingdon, 29-Dec-2017.)

Theoremdral1 1634 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 1635 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 1636 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 1637 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremdrnf2 1638 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremspimth 1639 Closed theorem form of spim 1642. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)

Theoremspimt 1640 Closed theorem form of spim 1642. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)

Theoremspimh 1641 Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1642 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 1642 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1642 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 1643 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 1644 Deduction version of spime 1645. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)

Theoremspime 1645 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 1646 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)

Theoremcbv3h 1647 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 1648 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 1649 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)

Theoremcbv2h 1650 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremcbv2 1651 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 1652 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremcbval 1653 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremcbvexh 1654 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)

Theoremcbvex 1655 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremchvar 1656 Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremequvini 1657 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 1658 A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1657.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)

Theoremnfald 1659 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 1660 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 1661 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 1662 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 1674.

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 1737, sbcom2 1879 and sbid2v 1888).

Note that our definition is valid even when and are replaced with the same variable, as sbid 1673 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 1883 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 1886.

When and are distinct, we can express proper substitution with the simpler expressions of sb5 1783 and sb6 1782.

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 1663 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)

Theoremsbbii 1664 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremsb1 1665 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsb2 1666 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ1 1667 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ2 1668 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremstdpc7 1669 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1607.) 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 1670 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ12r 1671 An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremsbequ12a 1672 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbid 1673 An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremstdpc4 1674 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 1675 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 1676 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 1677 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)

Theoremsb6x 1678 Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremnfs1f 1679 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbs1f 1680 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 1681 Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.)

Theoremsbequ6 1682 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.)

Theoremsbt 1683 A substitution into a theorem remains true. (See chvar 1656 and chvarv 1828 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremequsb1 1684 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremequsb2 1685 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremsbiedh 1686 Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1689). New proofs should use sbied 1687 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)

Theoremsbied 1687 Conversion of implicit substitution to explicit substitution (deduction version of sbie 1690). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsbiedv 1688* Conversion of implicit substitution to explicit substitution (deduction version of sbie 1690). (Contributed by NM, 7-Jan-2017.)

Theoremsbieh 1689 Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1690 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)

Theoremsbie 1690 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 1691 A property related to substitution that unlike equs5 1726 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremequs5e 1692 A property related to substitution that unlike equs5 1726 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)

Theoremax11e 1693 Analogue to ax-11 1413 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.)

Theoremax10oe 1694 Quantifier Substitution for existential quantifiers. Analogue to ax10o 1619 but for rather than . (Contributed by Jim Kingdon, 21-Dec-2017.)

Theoremdrex1 1695 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 1696 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 1697 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 1698 A version of sb4 1729 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremequs45f 1699 Two ways of expressing substitution when is not free in . (Contributed by NM, 25-Apr-2008.)

Theoremsb6f 1700 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.)

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