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Theorem List for Intuitionistic Logic Explorer - 1701-1800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremspim 1701 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1701 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.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremspimeh 1702 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.)
 |-  ( ph  ->  A. x ph )   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
Theoremspimed 1703 Deduction version of spime 1704. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)
 |-  ( ch  ->  F/ x ph )   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( ch  ->  (
 ph  ->  E. x ps )
 )
 
Theoremspime 1704 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.)
 |- 
 F/ x ph   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
Theoremcbv3 1705 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremcbv3h 1706 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.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremcbv1 1707 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.)
 |- 
 F/ x ph   &    |-  F/ y ph   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )   =>    |-  ( ph  ->  ( A. x ps  ->  A. y ch ) )
 
Theoremcbv1h 1708 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)
 |-  ( ph  ->  ( ps  ->  A. y ps )
 )   &    |-  ( ph  ->  ( ch  ->  A. x ch )
 )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch )
 ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch ) )
 
Theoremcbv2h 1709 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  A. y ps )
 )   &    |-  ( ph  ->  ( ch  ->  A. x ch )
 )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps 
 <-> 
 A. y ch )
 )
 
Theoremcbv2 1710 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.)
 |- 
 F/ x ph   &    |-  F/ y ph   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. y ch )
 )
 
Theoremcbvalh 1711 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
Theoremcbval 1712 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
Theoremcbvexh 1713 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
Theoremcbvex 1714 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
Theoremchvar 1715 Implicit substitution of  y for  x into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremequvini 1716 A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require  z to be distinct from  x and  y (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( x  =  y 
 ->  E. z ( x  =  z  /\  z  =  y ) )
 
Theoremequveli 1717 A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1716.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)
 |-  ( A. z ( z  =  x  <->  z  =  y
 )  ->  x  =  y )
 
Theoremnfald 1718 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y ps )
 
Theoremnfexd 1719 If  x is not free in  ph, it is not free in  E. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 7-Feb-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y ps )
 
1.3.10  Substitution (without distinct variables)
 
Syntaxwsb 1720 Extend wff definition to include proper substitution (read "the wff that results when  y is properly substituted for  x in wff  ph"). (Contributed by NM, 24-Jan-2006.)
 wff  [ y  /  x ] ph
 
Definitiondf-sb 1721 Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use  [ y  /  x ] ph to mean "the wff that results when  y is properly substituted for  x in the wff  ph." We can also use  [ y  /  x ] ph in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 1733.

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, " ph ( y ) is the wff that results when  y is properly substituted for  x in  ph ( x )." For example, if the original  ph ( x ) is  x  =  y, then  ph ( y ) is  y  =  y, from which we obtain that  ph ( x ) is  x  =  x. So what exactly does  ph ( x ) 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 1796, sbcom2 1940 and sbid2v 1949).

Note that our definition is valid even when  x and  y are replaced with the same variable, as sbid 1732 shows. We achieve this by having  x 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 1944 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 1947.

When  x and  y are distinct, we can express proper substitution with the simpler expressions of sb5 1843 and sb6 1842.

In classical logic, another possible definition is  ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph ) 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 
ph. (Contributed by NM, 5-Aug-1993.)

 |-  ( [ y  /  x ] ph  <->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
 ) )
 
Theoremsbimi 1722 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
 |-  ( ph  ->  ps )   =>    |-  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
 
Theoremsbbii 1723 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   =>    |-  ( [ y  /  x ] ph  <->  [ y  /  x ] ps )
 
Theoremsb1 1724 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
 
Theoremsb2 1725 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
 
Theoremsbequ1 1726 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  ->  [ y  /  x ] ph )
 )
 
Theoremsbequ2 1727 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( [ y  /  x ] ph  ->  ph )
 )
 
Theoremstdpc7 1728 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1664.) Translated to traditional notation, it can be read: " x  =  y  ->  ( ph ( x,  x )  ->  ph ( x,  y ) ), provided that  y is free for  x in  ph ( x,  y )." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
 |-  ( x  =  y 
 ->  ( [ x  /  y ] ph  ->  ph )
 )
 
Theoremsbequ12 1729 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  <->  [ y  /  x ] ph ) )
 
Theoremsbequ12r 1730 An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 |-  ( x  =  y 
 ->  ( [ x  /  y ] ph  <->  ph ) )
 
Theoremsbequ12a 1731 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( [ y  /  x ] ph  <->  [ x  /  y ] ph ) )
 
Theoremsbid 1732 An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( [ x  /  x ] ph  <->  ph )
 
Theoremstdpc4 1733 The specialization axiom of standard predicate calculus. It states that if a statement  ph holds for all  x, then it also holds for the specific case of  y (properly) substituted for  x. Translated to traditional notation, it can be read: " A. x ph ( x )  ->  ph ( y ), provided that  y is free for  x in  ph (
x )." Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  [ y  /  x ] ph )
 
Theoremsbh 1734 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.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( [ y  /  x ] ph  <->  ph )
 
Theoremsbf 1735 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.)
 |- 
 F/ x ph   =>    |-  ( [ y  /  x ] ph  <->  ph )
 
Theoremsbf2 1736 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
 |-  ( [ y  /  x ] A. x ph  <->  A. x ph )
 
Theoremsb6x 1737 Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremnfs1f 1738 If  x is not free in  ph, it is not free in  [ y  /  x ] ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |- 
 F/ x [ y  /  x ] ph
 
Theoremhbs1f 1739 If  x is not free in  ph, it is not free in  [ y  /  x ] ph. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
 
Theoremsbequ5 1740 Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.)
 |-  ( [ w  /  z ] A. x  x  =  y  <->  A. x  x  =  y )
 
Theoremsbequ6 1741 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.)
 |-  ( [ w  /  z ]  -.  A. x  x  =  y  <->  -.  A. x  x  =  y )
 
Theoremsbt 1742 A substitution into a theorem remains true. (See chvar 1715 and chvarv 1889 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ph   =>    |- 
 [ y  /  x ] ph
 
Theoremequsb1 1743 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |- 
 [ y  /  x ] x  =  y
 
Theoremequsb2 1744 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |- 
 [ y  /  x ] y  =  x
 
Theoremsbiedh 1745 Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1748). New proofs should use sbied 1746 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ch  ->  A. x ch ) )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( [
 y  /  x ] ps 
 <->  ch ) )
 
Theoremsbied 1746 Conversion of implicit substitution to explicit substitution (deduction version of sbie 1749). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch ) )
 
Theoremsbiedv 1747* Conversion of implicit substitution to explicit substitution (deduction version of sbie 1749). (Contributed by NM, 7-Jan-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch ) )
 
Theoremsbieh 1748 Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1749 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( [ y  /  x ] ph  <->  ps )
 
Theoremsbie 1749 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.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( [ y  /  x ] ph  <->  ps )
 
1.3.11  Theorems using axiom ax-11
 
Theoremequs5a 1750 A property related to substitution that unlike equs5 1785 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
 |-  ( E. x ( x  =  y  /\  A. y ph )  ->  A. x ( x  =  y  ->  ph ) )
 
Theoremequs5e 1751 A property related to substitution that unlike equs5 1785 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)
 |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
 
Theoremax11e 1752 Analogue to ax-11 1469 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.)
 |-  ( x  =  y 
 ->  ( E. x ( x  =  y  /\  ph )  ->  E. y ph ) )
 
Theoremax10oe 1753 Quantifier Substitution for existential quantifiers. Analogue to ax10o 1678 but for  E. rather than  A.. (Contributed by Jim Kingdon, 21-Dec-2017.)
 |-  ( A. x  x  =  y  ->  ( E. x ps  ->  E. y ps ) )
 
Theoremdrex1 1754 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.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. y ps )
 )
 
Theoremdrsb1 1755 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.)
 |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )
 
Theoremexdistrfor 1756 Distribution of existential quantifiers, with a bound-variable hypothesis saying that  y is not free in  ph, but  x can be free in  ph (and there is no distinct variable condition on  x and  y). (Contributed by Jim Kingdon, 25-Feb-2018.)
 |-  ( A. x  x  =  y  \/  A. x F/ y ph )   =>    |-  ( E. x E. y (
 ph  /\  ps )  ->  E. x ( ph  /\ 
 E. y ps )
 )
 
Theoremsb4a 1757 A version of sb4 1788 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] A. y ph  ->  A. x ( x  =  y  ->  ph )
 )
 
Theoremequs45f 1758 Two ways of expressing substitution when  y is not free in  ph. (Contributed by NM, 25-Apr-2008.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb6f 1759 Equivalence for substitution when  y is not free in  ph. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb5f 1760 Equivalence for substitution when  y is not free in  ph. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )
 )
 
Theoremsb4e 1761 One direction of a simplified definition of substitution that unlike sb4 1788 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  E. y ph )
 )
 
Theoremhbsb2a 1762 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] A. y ph  ->  A. x [ y  /  x ] ph )
 
Theoremhbsb2e 1763 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] E. y ph )
 
Theoremhbsb3 1764 If  y is not free in  ph,  x is not free in  [
y  /  x ] ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
 
Theoremnfs1 1765 If  y is not free in  ph,  x is not free in  [
y  /  x ] ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ y ph   =>    |- 
 F/ x [ y  /  x ] ph
 
Theoremsbcof2 1766 Version of sbco 1919 where  x is not free in  ph. (Contributed by Jim Kingdon, 28-Dec-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
 
1.4  Predicate calculus with distinct variables
 
1.4.1  Derive the axiom of distinct variables ax-16
 
Theoremspimv 1767* A version of spim 1701 with a distinct variable requirement instead of a bound-variable hypothesis. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( A. x ph  ->  ps )
 
Theoremaev 1768* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1770. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 |-  ( A. x  x  =  y  ->  A. z  w  =  v )
 
Theoremax16 1769* Theorem showing that ax-16 1770 is redundant if ax-17 1491 is included in the axiom system. The important part of the proof is provided by aev 1768.

See ax16ALT 1815 for an alternate proof that does not require ax-10 1468 or ax-12 1474.

This theorem should not be referenced in any proof. Instead, use ax-16 1770 below so that theorems needing ax-16 1770 can be more easily identified. (Contributed by NM, 8-Nov-2006.)

 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Axiomax-16 1770* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1491 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 1491; see theorem ax16 1769.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1769. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theoremdveeq2 1771* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y ) )
 
Theoremdveeq2or 1772* Quantifier introduction when one pair of variables is distinct. Like dveeq2 1771 but connecting  A. x x  =  y by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.)
 |-  ( A. x  x  =  y  \/  F/ x  z  =  y
 )
 
TheoremdvelimfALT2 1773* Proof of dvelimf 1968 using dveeq2 1771 (shown as the last hypothesis) instead of ax-12 1474. This shows that ax-12 1474 could be replaced by dveeq2 1771 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   &    |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
 )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremnd5 1774* A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.)
 |-  ( -.  A. y  y  =  x  ->  ( z  =  y  ->  A. x  z  =  y ) )
 
Theoremexlimdv 1775* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  ch ) )
 
Theoremax11v2 1776* Recovery of ax11o 1778 from ax11v 1783 without using ax-11 1469. The hypothesis is even weaker than ax11v 1783, with  z both distinct from  x and not occurring in  ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1778. (Contributed by NM, 2-Feb-2007.)
 |-  ( x  =  z 
 ->  ( ph  ->  A. x ( x  =  z  -> 
 ph ) ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) ) )
 
Theoremax11a2 1777* Derive ax-11o 1779 from a hypothesis in the form of ax-11 1469. The hypothesis is even weaker than ax-11 1469, with  z both distinct from  x and not occurring in  ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1778. (Contributed by NM, 2-Feb-2007.)
 |-  ( x  =  z 
 ->  ( A. z ph  ->  A. x ( x  =  z  ->  ph )
 ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
1.4.2  Derive the obsolete axiom of variable substitution ax-11o
 
Theoremax11o 1778 Derivation of set.mm's original ax-11o 1779 from the shorter ax-11 1469 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1770 or ax-17 1491.

Normally, ax11o 1778 should be used rather than ax-11o 1779, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
Axiomax-11o 1779 Axiom ax-11o 1779 ("o" for "old") was the original version of ax-11 1469, before it was discovered (in Jan. 2007) that the shorter ax-11 1469 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 " -.  A. x x  =  y  ->..." as informally meaning "if  x and  y are distinct variables, then..." The antecedent becomes false if the same variable is substituted for  x and  y, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form  -.  A. x x  =  y a "distinctor."

This axiom is redundant, as shown by theorem ax11o 1778.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1778. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
1.4.3  More theorems related to ax-11 and substitution
 
Theoremalbidv 1780* Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbidv 1781* Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theoremax11b 1782 A bidirectional version of ax-11o 1779. (Contributed by NM, 30-Jun-2006.)
 |-  ( ( -.  A. x  x  =  y  /\  x  =  y
 )  ->  ( ph  <->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremax11v 1783* This is a version of ax-11o 1779 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)
 |-  ( x  =  y 
 ->  ( ph  ->  A. x ( x  =  y  -> 
 ph ) ) )
 
Theoremax11ev 1784* Analogue to ax11v 1783 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
 |-  ( x  =  y 
 ->  ( E. x ( x  =  y  /\  ph )  ->  ph ) )
 
Theoremequs5 1785 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremequs5or 1786 Lemma used in proofs of substitution properties. Like equs5 1785 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.)
 |-  ( A. x  x  =  y  \/  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremsb3 1787 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
 
Theoremsb4 1788 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremsb4or 1789 One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1788 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)
 |-  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremsb4b 1790 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
 |-  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremsb4bor 1791 Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)
 |-  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremhbsb2 1792 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  A. x [
 y  /  x ] ph ) )
 
Theoremnfsb2or 1793 Bound-variable hypothesis builder for substitution. Similar to hbsb2 1792 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.)
 |-  ( A. x  x  =  y  \/  F/ x [ y  /  x ] ph )
 
Theoremsbequilem 1794 Propositional logic lemma used in the sbequi 1795 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
 |-  ( ph  \/  ( ps  ->  ( ch  ->  th ) ) )   &    |-  ( ta  \/  ( ps  ->  ( th  ->  et )
 ) )   =>    |-  ( ph  \/  ( ta  \/  ( ps  ->  ( ch  ->  et )
 ) ) )
 
Theoremsbequi 1795 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.)
 |-  ( x  =  y 
 ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph )
 )
 
Theoremsbequ 1796 An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )
 
Theoremdrsb2 1797 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.)
 |-  ( A. x  x  =  y  ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )
 
Theoremspsbe 1798 A specialization theorem, mostly the same as Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 29-Dec-2017.)
 |-  ( [ y  /  x ] ph  ->  E. x ph )
 
Theoremspsbim 1799 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
 )
 
Theoremspsbbi 1800 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps ) )
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