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Theorem List for Intuitionistic Logic Explorer - 1501-1600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhbnaes 1501 Rule that applies hbnae 1499 to antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremnaecoms 1502 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)

Theoremequs4 1503 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)

Theoremequsal 1504 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremequsex 1505 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theoremequsexd 1506 Deduction form of equsex 1505. (Contributed by Jim Kingdon, 29-Dec-2017.)

Theoremdral1 1507 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 1508 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 1509 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 1510 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremdrnf2 1511 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theorema4imt 1512 Closed theorem form of a4im 1513. (Contributed by NM, 15-Jan-2008.)

Theorema4im 1513 Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The a4im 1513 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.)

Theorema4ime 1514 Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.)

Theorema4imed 1515 Deduction version of a4ime 1514. (Contributed by NM, 5-Aug-1993.)

Theoremcbv1 1516 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)

Theoremcbv2 1517 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)

Theoremcbv3h 1518 Rule used to change bound variables, using implicit substitition, that does not use ax-12 1335. (Contributed by NM, 5-Aug-1993.)

Theoremcbv3ALT 1519 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremcbvalh 1520 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremcbval 1521 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremcbvexh 1522 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)

Theoremcbvex 1523 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremchvar 1524 Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.)

Theoremequvini 1525 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 1526 A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1525.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)

Theoremnfald 1527 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 1528 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.)

Theoremnfa2 1529 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfia1 1530 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)

1.3.10  Substitution (without distinct variables)

Syntaxwsbc 1531 Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class for set variable in wff ."

(The purpose of introducing here is to allow us to express i.e. "prove" the wsb 1532 of predicate calculus in terms of the wsbc 1531 of set theory, so that we don't "overload" its connectives with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variable is introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus.)

Theoremwsb 1532 Extend wff definition to include proper substitution (read "the wff that results when is properly substituted for in wff ").

(Instead of introducing wsb 1532 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 wsbc 1531. This lets us avoid overloading its connectives, thus preventing ambiguity that would complicate some Metamath parsers. Note: To see the proof steps of this syntax proof, type "show proof wsb /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

Definitiondf-sb 1533 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 1545.

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 1607, sbcom2 1739 and sbid2v 1749).

Note that our definition is valid even when and are replaced with the same variable, as sbid 1544 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 1744 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 1747.

When and are distinct, we can express proper substitution with the simpler expressions of sb5 1649 and sb6 1648.

In classical logic, we have additional equivalent definitions dfsb2 2184 and dfsb3 2185, but we do not have intuitionistic proofs that those are equivalent.

There are no restrictions on any of the variables, including what variables may occur in wff . (Contributed by NM, 5-Aug-1993.)

Theoremsbimi 1534 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)

Theoremsbbii 1535 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremsb1 1536 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsb2 1537 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ1 1538 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ2 1539 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremstdpc7 1540 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1481.) 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 1541 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ12r 1542 An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremsbequ12a 1543 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbid 1544 An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremstdpc4 1545 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 1546 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 1547 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 1548 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)

Theoremsb6x 1549 Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremnfs1f 1550 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbs1f 1551 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 1552 Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.)

Theoremsbequ6 1553 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.)

Theoremsbt 1554 A substitution into a theorem remains true. (See chvar 1524 and chvarv 1688 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremequsb1 1555 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremequsb2 1556 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremsbiedh 1557 Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1559). (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsbied 1558 Conversion of implicit substitution to explicit substitution (deduction version of sbie 1560). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsbieh 1559 Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.)

Theoremsbie 1560 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 1561 A property related to substitution that unlike equs5 1596 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremequs5e 1562 A property related to substitution that unlike equs5 1596 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)

Theoremax11e 1563 Analogue to ax-11 1330 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.)

Theoremax10oe 1564 Quantifier Substitution for existential quantifiers. Analogue to ax10o 1493 but for rather than . (Contributed by Jim Kingdon, 21-Dec-2017.)

Theoremdrex1 1565 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 1566 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.)

Theoremexdistrf 1567 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 Mario Carneiro, 20-Mar-2013.) (Hypothesis and proof modified for intuitionistic logic by Jim Kingdon, 25-Feb-2018.)

Theoremsb4a 1568 A version of sb4 1599 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremequs45f 1569 Two ways of expressing substitution when is not free in . (Contributed by NM, 25-Apr-2008.)

Theoremsb6f 1570 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.)

Theoremsb5f 1571 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.)

Theoremsb4e 1572 One direction of a simplified definition of substitution that unlike sb4 1599 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb2a 1573 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb2e 1574 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb3 1575 If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.)

Theoremnfs1 1576 If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremsbcof2 1577 Version of sbco 1719 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

Theorema4imv 1578* A version of a4im 1513 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.)

Theoremaev 1579* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1581. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremax16 1580* Theorem showing that ax-16 1581 is redundant if ax-17 1350 is included in the axiom system. The important part of the proof is provided by aev 1579.

See ax16ALT 1622 for an alternate proof that does not require ax-10 1329 or ax-12 1335.

This theorem should not be referenced in any proof. Instead, use ax-16 1581 below so that theorems needing ax-16 1581 can be more easily identified. (Contributed by NM, 8-Nov-2006.)

Axiomax-16 1581* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1350 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 1350; see theorem ax16 1580.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1580. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremdveeq2 1582* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Theoremdveeq2or 1583* Quantifier introduction when one pair of variables is distinct. Like dveeq2 1582 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 1584* Proof of dvelimf 1766 using dveeq2 1582 (shown as the last hypothesis) instead of ax-12 1335. This shows that ax-12 1335 could be replaced by dveeq2 1582 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.)

Theoremnd5 1585* A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.)

Theoremexlimdv 1586* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)

Theoremax11v2 1587* Recovery of ax11o 1589 from ax11v 1594 without using ax-11 1330. The hypothesis is even weaker than ax11v 1594, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1589. (Contributed by NM, 2-Feb-2007.)

Theoremax11a2 1588* Derive ax-11o 1590 from a hypothesis in the form of ax-11 1330. The hypothesis is even weaker than ax-11 1330, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1589. (Contributed by NM, 2-Feb-2007.)

1.4.2  Derive the obsolete axiom of variable substitution ax-11o

Theoremax11o 1589 Derivation of set.mm's original ax-11o 1590 from the shorter ax-11 1330 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1581 or ax-17 1350.

Normally, ax11o 1589 should be used rather than ax-11o 1590, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

Axiomax-11o 1590 Axiom ax-11o 1590 ("o" for "old") was the original version of ax-11 1330, before it was discovered (in Jan. 2007) that the shorter ax-11 1330 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). 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. To understand this theorem more easily, think of " ..." as informally meaning "if and are distinct variables then..." The antecedent becomes false if the same variable is substituted for and , ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form a "distinctor."

This axiom is redundant, as shown by theorem ax11o 1589.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1589. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

1.4.3  More theorems related to ax-11 and substitution

Theoremalbidv 1591* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)

Theoremexbidv 1592* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)

Theoremax11b 1593 A bidirectional version of ax-11o 1590. (Contributed by NM, 30-Jun-2006.)

Theoremax11v 1594* This is a version of ax-11o 1590 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)

Theoremax11ev 1595* Analogue to ax11v 1594 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)

Theoremequs5 1596 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

Theoremequs5or 1597 Lemma used in proofs of substitution properties. Like equs5 1596 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.)

Theoremsb3 1598 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)

Theoremsb4 1599 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)

Theoremsb4or 1600 One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1599 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)

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