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Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbi2NEW7 29501 Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)
 |-  (
 ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
 
TheoremsbimNEW7 29502 Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
TheoremsborNEW7 29503 Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.)
 |-  ( [ y  /  x ] ( ph  \/  ps )  <->  ( [ y  /  x ] ph  \/  [ y  /  x ] ps ) )
 
TheoremsbrimNEW7 29504 Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  F/ x ph   =>    |-  ( [ y  /  x ] ( ph  ->  ps )  <->  ( ph  ->  [ y  /  x ] ps ) )
 
TheoremsblimNEW7 29505 Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  F/ x ps   =>    |-  ( [ y  /  x ] ( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps ) )
 
TheoremsbanNEW7 29506 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  /\  ps ) 
 <->  ( [ y  /  x ] ph  /\  [
 y  /  x ] ps ) )
 
TheoremsbbiNEW7 29507 Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  <->  ps )  <->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps ) )
 
TheoremspsbeNEW7 29508 A specialization theorem. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  E. x ph )
 
TheoremspsbimNEW7 29509 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A. x ( ph  ->  ps )  ->  ( [
 y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
TheoremspsbbiNEW7 29510 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ( ph  <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps ) )
 
TheoremsbbidNEW7 29511 Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  [ y  /  x ] ch ) )
 
Theoremsbequ8NEW7 29512 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  [ y  /  x ] ( x  =  y  ->  ph ) )
 
Theoremnfsb4twAUX7 29513* Weak version of nfsb4t 2154 not requiring ax-7 1749. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )
 
Theoremnfsb4wAUX7 29514* Weak version of nfsb4t 2154 not requiring ax-7 1749. (Contributed by NM, 27-Oct-2017.)
 |-  F/ z ph   =>    |-  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph )
 
TheoremnfsbdwAUX7 29515* Deduction version of nfsbwAUX7 29602. (Contributed by NM, 24-May-2018.)
 |-  F/ x ph   &    |-  ( ph  ->  F/ z ps )   =>    |-  ( ph  ->  F/ z [ y  /  x ] ps )
 
TheoremsbequiNEW7 29516 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph )
 )
 
TheoremsbequNEW7 29517 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 ) )
 
Theoremdrsb2NEW7 29518 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 ) )
 
TheoremsbcoNEW7 29519 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
 
Theoremsbid2NEW7 29520 An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ x ph   =>    |-  ( [ y  /  x ] [ x  /  y ] ph  <->  ph )
 
TheoremsbidmNEW7 29521 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( [ y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph )
 
Theoremsbco2wAUX7 29522* Weak version of sbco2 2161 not requiring ax-7 1749. (Contributed by NM, 27-Oct-2017.)
 |-  F/ z ph   =>    |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
 
Theoremsbco2dwAUX7 29523* Weak version of sbco2d 2162 not requiring ax-7 1749. (Contributed by NM, 27-Oct-2017.)
 |-  F/ x ph   &    |-  F/ z ph   &    |-  ( ph  ->  F/ z ps )   =>    |-  ( ph  ->  ( [ y  /  z ] [ z  /  x ] ps  <->  [ y  /  x ] ps ) )
 
Theoremsbco3wAUX7 29524* Weak version of sbco3 2163 not requiring ax-7 1749. (Contributed by NM, 27-Oct-2017.)
 |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
 
TheoremsbcomwAUX7 29525* Weak version of sbcom 2164 not requiring ax-7 1749. (Contributed by NM, 27-Oct-2017.)
 |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
 
Theoremsb8iwAUX7 29526* Version of sb8 2167 not requiring ax-7 1749. (Contributed by NM, 22-May-2018.)
 |-  ( -.  A. x  x  =  y  ->  F/ y ps )   =>    |-  ( -.  A. x  x  =  y  ->  (
 A. y [ y  /  x ] ps  ->  A. x ps ) )
 
Theoremsb8dwAUX7 29527* Weak version of sb8 2167 not requiring ax-7 1749. (Contributed by NM, 22-May-2018.)
 |-  F/ x ph   &    |-  F/ y ph   &    |-  ( ph  ->  F/ y ps )   =>    |-  ( ph  ->  ( A. x ps  <->  A. y [ y  /  x ] ps )
 )
 
Theoremsb5rfNEW7 29528 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ y ph   =>    |-  ( ph  <->  E. y ( y  =  x  /\  [
 y  /  x ] ph ) )
 
Theoremsb6rfNEW7 29529 Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ y ph   =>    |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
 )
 
Theoremsb8wAUX7 29530* Weak version of sb8 2167 not requiring ax-7 1749. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   =>    |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
 
Theoremsb8ewAUX7 29531* Weak version of sb8e 2168 not requiring ax-7 1749. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   =>    |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
 
Theoremax11vNEW7 29532* This is a version of ax-11o 2217 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 2074 for the rederivation of ax-11o 2217 from this theorem. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremsb56NEW7 29533* Two equivalent ways of expressing the proper substitution of  y for  x in  ph, when  x and  y are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1659. (Contributed by NM, 14-Apr-2008.)
 |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
 
Theoremsb6NEW7 29534* Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb5NEW7 29535* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )
 )
 
TheoremexsbNEW7 29536* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) (Revised by NM, 28-Nov-2017.) Revised to prove from ax-7v 29379 instead of ax-7 1749.
 |-  ( E. x ph  <->  E. y A. x ( x  =  y  -> 
 ph ) )
 
TheoremexsbOLDNEW7 29537* An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (New usage is discouraged.)
 |-  ( E. x ph  <->  E. y A. x ( x  =  y  -> 
 ph ) )
 
Theoremequsb3lemNEW7 29538* Lemma for equsb3 2177. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
 
Theoremequsb3NEW7 29539* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
 |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
 
Theoremelsb3NEW7 29540* Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  e.  z  <->  x  e.  z )
 
Theoremelsb4NEW7 29541* Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] z  e.  y  <->  z  e.  x )
 
Theoremhbs1NEW7 29542*  x is not free in  [ y  /  x ] ph when  x and  y are distinct. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  A. x [
 y  /  x ] ph )
 
Theoremnfs1vNEW7 29543*  x is not free in  [ y  /  x ] ph when  x and  y are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/ x [ y  /  x ] ph
 
TheoremsbhbwAUX7 29544* Weak version of sbhb 2182 not requiring ax-7 1749. (Contributed by NM, 28-Oct-2017.)
 |-  (
 ( ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) )
 
Theoremsbnf2NEW7 29545* Two ways of expressing " x is (effectively) not free in  ph." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  ( F/ x ph  <->  A. y A. z
 ( [ y  /  x ] ph  <->  [ z  /  x ] ph ) )
 
Theorem2sb5NEW7 29546* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  <->  E. x E. y
 ( ( x  =  z  /\  y  =  w )  /\  ph )
 )
 
Theorem2sb6NEW7 29547* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  <->  A. x A. y
 ( ( x  =  z  /\  y  =  w )  ->  ph )
 )
 
Theoremsb6aNEW7 29548* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  [ x  /  y ] ph )
 )
 
Theoremsbid2vNEW7 29549* An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] [ x  /  y ] ph  <->  ph )
 
TheoremsbelxNEW7 29550* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph 
 <-> 
 E. x ( x  =  y  /\  [ x  /  y ] ph ) )
 
Theoremsbel2xNEW7 29551* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph 
 <-> 
 E. x E. y
 ( ( x  =  z  /\  y  =  w )  /\  [
 y  /  w ] [ x  /  z ] ph ) )
 
Theoremsbal1NEW7 29552* A theorem used in elimination of disjoint variable restriction on  x and  y by replacing it with a distinctor  -.  A. x x  =  z. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  z  ->  ( [
 z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
 )
 
TheoremsbalNEW7 29553* Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
 
TheoremsbexNEW7 29554* Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
 |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
 
TheoremsbalvNEW7 29555* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  ps )   =>    |-  ( [ y  /  x ] A. z ph  <->  A. z ps )
 
TheoremnaecomsNEW7 29556 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. y  y  =  x  -> 
 ph )
 
TheoremchvarNEW7 29557 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
 
Theoremequs45fNEW7 29558 Two ways of expressing substitution when  y is not free in  ph. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  F/ y ph   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremax11bNEW7 29559 A bidirectional version of ax11oNEW7 29469. (Contributed by NM, 30-Jun-2006.)
 |-  (
 ( -.  A. x  x  =  y  /\  x  =  y )  ->  ( ph  <->  A. x ( x  =  y  ->  ph )
 ) )
 
TheoremspvNEW7 29560* Specialization, using implicit substitution. (Contributed by NM, 30-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
TheoremspimevNEW7 29561* Distinct-variable version of spimeNEW7 29446. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
TheoremspeivNEW7 29562* Inference from existential specialization, using implicit substitution. (Contributed by NM, 19-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ps   =>    |-  E. x ph
 
TheoremchvarvNEW7 29563* Implicit substitution of  y for  x into a theorem. (Contributed by NM, 20-Apr-1994.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
TheoremcleljustNEW7 29564* When the class variables in definition df-clel 2431 are replaced with set variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the set variables in wel 1726 with the class variables in wcel 1725. Note: This proof is referenced on the Metamath Proof Explorer Home Page and shouldn't be changed. (Contributed by NM, 28-Jan-2004.) (Proof modification is discouraged.)
 |-  ( x  e.  y  <->  E. z ( z  =  x  /\  z  e.  y ) )
 
TheoremcleljustALTNEW7 29565* When the class variables in definition df-clel 2431 are replaced with set variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the set variables in wel 1726 with the class variables in wcel 1725. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.)
 |-  ( x  e.  y  <->  E. z ( z  =  x  /\  z  e.  y ) )
 
Theoremsb6xNEW7 29566 Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  F/ x ph   =>    |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremnfs1fNEW7 29567 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
 
TheoremsbtNEW7 29568 A substitution into a theorem remains true. (See chvarNEW7 29557 and chvarvNEW7 29563 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ph   =>    |- 
 [ y  /  x ] ph
 
TheoremsbiedvNEW7 29569* Conversion of implicit substitution to explicit substitution (deduction version of sbieNEW7 29478). (Contributed by NM, 7-Jan-2017.)
 |-  (
 ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch ) )
 
Theoremsb6fNEW7 29570 Equivalence for substitution when  y is not free in  ph. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  F/ y ph   =>    |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb5fNEW7 29571 Equivalence for substitution when  y is not free in  ph. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  F/ y ph   =>    |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )
 )
 
Theoremax16ALTNEW7 29572* Alternate proof of ax16NEW7 29486. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
TheoremdvelimALTNEW7 29573* Version of dvelim 2069 that doesn't use ax-10 2216. (See dvelimh 2067 for a version that doesn't use ax-11 1761.) (Contributed by NM, 17-May-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremsb3NEW7 29574 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 ) )
 
Theoremdfsb2NEW7 29575 An alternate definition of proper substitution that, like df-sb 1659, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.)
 |-  ( [ y  /  x ] ph  <->  ( ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremdfsb3NEW7 29576 An alternate definition of proper substitution df-sb 1659 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)
 |-  ( [ y  /  x ] ph  <->  ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremsb3anNEW7 29577 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)
 |-  ( [ y  /  x ] ( ph  /\  ps  /\ 
 ch )  <->  ( [ y  /  x ] ph  /\  [
 y  /  x ] ps  /\  [ y  /  x ] ch ) )
 
TheoremsblbisNEW7 29578 Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  ps )   =>    |-  ( [ y  /  x ] ( ch  <->  ph )  <->  ( [ y  /  x ] ch  <->  ps ) )
 
TheoremsbrbisNEW7 29579 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  ps )   =>    |-  ( [ y  /  x ] ( ph  <->  ch )  <->  ( ps  <->  [ y  /  x ] ch ) )
 
TheoremsbrbifNEW7 29580 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  F/ x ch   &    |-  ( [ y  /  x ] ph  <->  ps )   =>    |-  ( [ y  /  x ] ( ph  <->  ch )  <->  ( ps  <->  ch ) )
 
Theoremsbcom2NEW7 29581* Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.)
 |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
 
Theoremax7w1AUX7 29582 Weak version of ax-7 1749 not requiring ax-7 1749. (Contributed by NM, 9-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( A. x A. y ph  ->  A. y A. x ph ) )
 
Theoremax7w1hAUX7 29583 Weak version of hbal 1751 not requiring ax-7 1749. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )   =>    |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x A. y ph ) )
 
Theoremhbaew0AUX7 29584 Weak version of hbae 2040 not requiring ax-7 1749. (Contributed by NM, 29-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z  x  =  y )
 
Theoremhbaew4AUX7 29585 Weak version of hbae 2040 not requiring ax-7 1749. (Contributed by NM, 30-Oct-2017.)
 |-  ( A. x  x  =  z  ->  ( A. x  x  =  y  ->  A. z A. x  x  =  y ) )
 
Theoremhbaew5AUX7 29586* Weak version of hbae 2040 not requiring ax-7 1749. (Contributed by NM, 30-Oct-2017.)
 |-  ( A. u  u  =  v  ->  ( A. x  x  =  y  ->  A. z A. x  x  =  y ) )
 
Theoremax7w2AUX7 29587 Special case of ax-7 1749. (Contributed by NM, 9-Oct-2017.)
 |-  ( A. x A. y [
 y  /  x ] ph  ->  A. y A. x [ y  /  x ] ph )
 
Theoremax7w3AUX7 29588 Special case of ax-7 1749. (Contributed by NM, 12-Oct-2017.)
 |-  ( A. x A. y  x  =  y  ->  A. y A. x  x  =  y )
 
Theoremax7w6AUX7 29589 Version of sb9i 2169 with the usage of ax-7 1749 broken out as a hypothesis. (Contributed by NM, 16-Oct-2017.)
 |-  ( A. x A. y [ x  /  y ] ph  ->  A. y A. x [ x  /  y ] ph )   =>    |-  ( A. x [ x  /  y ] ph  ->  A. y [ y  /  x ] ph )
 
Theoremax7w7AUX7 29590 Special case of ax-7 1749. (Contributed by NM, 12-Oct-2017.)
 |-  ( A. x A. y  -.  x  =  y  ->  A. y A. x  -.  x  =  y )
 
Theoremax7w8AUX7 29591 Special case of ax-7 1749. (Contributed by NM, 13-Oct-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax7nfAUX7 29592 Special case of ax-7 1749. (Contributed by NM, 23-Nov-2017.)
 |-  F/ x ph   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax7w7tAUX7 29593 Special case of ax-7 1749. (Contributed by NM, 23-Nov-2017.)
 |-  ( A. y ( ph  ->  A. x ph )  ->  ( A. x A. y ph  ->  A. y A. x ph ) )
 
Theoremax7wnftAUX7 29594 Special case of ax-7 1749. (Contributed by NM, 23-Nov-2017.)
 |-  ( A. y F/ x ph  ->  ( A. x A. y ph  ->  A. y A. x ph ) )
 
Theoremax7w4AUX7 29595 Remove quantifier from ax-7 1749. (Contributed by NM, 12-Oct-2017.)
 |-  ( A. x A. y A. y ph  ->  A. y A. x A. y ph )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax7w5AUX7 29596 Same as hbal 1751 with explicit ax-7 1749 hypothesis. (Contributed by NM, 12-Oct-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( A. y A. x ph  ->  A. x A. y ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
Theoremax7w9AUX7 29597 Special case of ax-7 1749 proved from ax-7v 29379. (Contributed by NM, 28-Nov-2017.)
 |-  ( A. x A. y ( x  =  y  /\  x  =  z )  ->  A. y A. x ( x  =  y  /\  x  =  z
 ) )
 
Theoremalcomw9bAUX7 29598 Special case of alcom 1752 proved from ax-7v 29379. (Contributed by NM, 28-Nov-2017.)
 |-  ( A. x A. y ( x  =  y  /\  x  =  z )  <->  A. y A. x ( x  =  y  /\  x  =  z )
 )
 
Theoremax7w10AUX7 29599 Special case of ax-7 1749 derived from ax-7v 29379. (Contributed by NM, 20-May-2018.)
 |-  ( A. x A. y ( x  =  y  /\  ph )  ->  A. y A. x ( x  =  y  /\  ph )
 )
 
Theoremax7w11AUX7 29600 Special case of ax-7 1749 derived from ax-7v 29379. (Contributed by NM, 20-May-2018.)
 |-  ( A. x A. y ( -.  x  =  y 
 /\  ph )  ->  A. y A. x ( -.  x  =  y  /\  ph )
 )
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