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Theorem List for Metamath Proof Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrspcedv 2901* Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps ) )
 
Theoremrspc2 2902* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)
 |- 
 F/ x ch   &    |-  F/ y ps   &    |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  ( y  =  B  ->  ( ch  <->  ps ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. x  e.  C  A. y  e.  D  ph  ->  ps )
 )
 
Theoremrspc2v 2903* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
 |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  B  ->  ( ch  <->  ps ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. x  e.  C  A. y  e.  D  ph  ->  ps ) )
 
Theoremrspc2va 2904* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
 |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  B  ->  ( ch  <->  ps ) )   =>    |-  ( ( ( A  e.  C  /\  B  e.  D )  /\  A. x  e.  C  A. y  e.  D  ph )  ->  ps )
 
Theoremrspc2ev 2905* 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
 |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  B  ->  ( ch  <->  ps ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D  /\  ps )  ->  E. x  e.  C  E. y  e.  D  ph )
 
Theoremrspc3v 2906* 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
 |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  B  ->  ( ch  <->  th ) )   &    |-  (
 z  =  C  ->  ( th  <->  ps ) )   =>    |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T )  ->  ( A. x  e.  R  A. y  e.  S  A. z  e.  T  ph  ->  ps )
 )
 
Theoremrspc3ev 2907* 3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
 |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  B  ->  ( ch  <->  th ) )   &    |-  (
 z  =  C  ->  ( th  <->  ps ) )   =>    |-  ( ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T )  /\  ps )  ->  E. x  e.  R  E. y  e.  S  E. z  e.  T  ph )
 
Theoremeqvinc 2908* A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  A  e.  _V   =>    |-  ( A  =  B 
 <-> 
 E. x ( x  =  A  /\  x  =  B ) )
 
Theoremeqvincf 2909 A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  A  e.  _V   =>    |-  ( A  =  B  <->  E. x ( x  =  A  /\  x  =  B ) )
 
Theoremalexeq 2910* Two ways to express substitution of 
A for  x in  ph. (Contributed by NM, 2-Mar-1995.)
 |-  A  e.  _V   =>    |-  ( A. x ( x  =  A  -> 
 ph )  <->  E. x ( x  =  A  /\  ph )
 )
 
Theoremceqex 2911* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
 |-  ( x  =  A  ->  ( ph  <->  E. x ( x  =  A  /\  ph )
 ) )
 
Theoremceqsexg 2912* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
 
Theoremceqsexgv 2913* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
 
Theoremceqsrexv 2914* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  B  ->  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ps ) )
 
Theoremceqsrexbv 2915* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ( A  e.  B  /\  ps ) )
 
Theoremceqsrex2v 2916* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( E. x  e.  C  E. y  e.  D  ( ( x  =  A  /\  y  =  B )  /\  ph )  <->  ch ) )
 
Theoremclel2 2917* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
 |-  A  e.  _V   =>    |-  ( A  e.  B 
 <-> 
 A. x ( x  =  A  ->  x  e.  B ) )
 
Theoremclel3g 2918* An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
 |-  ( B  e.  V  ->  ( A  e.  B  <->  E. x ( x  =  B  /\  A  e.  x ) ) )
 
Theoremclel3 2919* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
 |-  B  e.  _V   =>    |-  ( A  e.  B 
 <-> 
 E. x ( x  =  B  /\  A  e.  x ) )
 
Theoremclel4 2920* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
 |-  B  e.  _V   =>    |-  ( A  e.  B 
 <-> 
 A. x ( x  =  B  ->  A  e.  x ) )
 
Theorempm13.183 2921* Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only  A is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( A  e.  V  ->  ( A  =  B  <->  A. z ( z  =  A  <->  z  =  B ) ) )
 
Theoremrr19.3v 2922* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the non-empty class condition of r19.3rzv 3560 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.)
 |-  ( A. x  e.  A  A. y  e.  A  ph  <->  A. x  e.  A  ph )
 
Theoremrr19.28v 2923* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the non-empty class condition of r19.28zv 3562 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)
 |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps ) 
 <-> 
 A. x  e.  A  ( ph  /\  A. y  e.  A  ps ) )
 
Theoremelabgt 2924* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2928.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  (
 ph 
 <->  ps ) ) ) 
 ->  ( A  e.  { x  |  ph }  <->  ps ) )
 
Theoremelabgf 2925 Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   &    |-  F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps ) )
 
Theoremelabf 2926* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |- 
 F/ x ps   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  { x  |  ph }  <->  ps )
 
Theoremelab 2927* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  { x  |  ph }  <->  ps )
 
Theoremelabg 2928* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps ) )
 
Theoremelab2g 2929* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  B  =  { x  |  ph }   =>    |-  ( A  e.  V  ->  ( A  e.  B  <->  ps ) )
 
Theoremelab2 2930* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  B  =  { x  |  ph }   =>    |-  ( A  e.  B  <->  ps )
 
Theoremelab4g 2931* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  B  =  { x  |  ph }   =>    |-  ( A  e.  B  <->  ( A  e.  _V  /\  ps ) )
 
Theoremelab3gf 2932 Membership in a class abstraction, with a weaker antecedent than elabgf 2925. (Contributed by NM, 6-Sep-2011.)
 |-  F/_ x A   &    |-  F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( ps 
 ->  A  e.  B ) 
 ->  ( A  e.  { x  |  ph }  <->  ps ) )
 
Theoremelab3g 2933* Membership in a class abstraction, with a weaker antecedent than elabg 2928. (Contributed by NM, 29-Aug-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( ps 
 ->  A  e.  B ) 
 ->  ( A  e.  { x  |  ph }  <->  ps ) )
 
Theoremelab3 2934* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
 |-  ( ps  ->  A  e.  _V )   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  { x  |  ph }  <->  ps )
 
Theoremelrabf 2935 Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  { x  e.  B  |  ph
 } 
 <->  ( A  e.  B  /\  ps ) )
 
Theoremelrab 2936* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  { x  e.  B  |  ph
 } 
 <->  ( A  e.  B  /\  ps ) )
 
Theoremelrab3 2937* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  B  ->  ( A  e.  { x  e.  B  |  ph
 } 
 <->  ps ) )
 
Theoremelrab2 2938* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  C  =  { x  e.  B  |  ph }   =>    |-  ( A  e.  C  <->  ( A  e.  B  /\  ps ) )
 
Theoremralab 2939* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( y  =  x 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  { y  |  ph } ch  <->  A. x ( ps 
 ->  ch ) )
 
Theoremralrab 2940* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( y  =  x 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  { y  e.  A  |  ph } ch  <->  A. x  e.  A  ( ps  ->  ch )
 )
 
Theoremrexab 2941* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  ( y  =  x 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( ps 
 /\  ch ) )
 
Theoremrexrab 2942* Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  ( y  =  x 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  { y  e.  A  |  ph } ch  <->  E. x  e.  A  ( ps  /\  ch )
 )
 
Theoremralab2 2943* Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( x  =  y 
 ->  ( ps  <->  ch ) )   =>    |-  ( A. x  e.  { y  |  ph } ps  <->  A. y ( ph  ->  ch ) )
 
Theoremralrab2 2944* Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( x  =  y 
 ->  ( ps  <->  ch ) )   =>    |-  ( A. x  e.  { y  e.  A  |  ph } ps  <->  A. y  e.  A  ( ph  ->  ch )
 )
 
Theoremrexab2 2945* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( x  =  y 
 ->  ( ps  <->  ch ) )   =>    |-  ( E. x  e.  { y  |  ph } ps  <->  E. y ( ph  /\ 
 ch ) )
 
Theoremrexrab2 2946* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( x  =  y 
 ->  ( ps  <->  ch ) )   =>    |-  ( E. x  e.  { y  e.  A  |  ph } ps  <->  E. y  e.  A  ( ph  /\  ch )
 )
 
Theoremabidnf 2947* Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)
 |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
 
Theoremdedhb 2948* A deduction theorem for converting the inference  |- 
F/_ x A =>  |-  ph into a closed theorem. Use nfa1 1768 and nfab 2436 to eliminate the hypothesis of the substitution instance  ps of the inference. For converting the inference form into a deduction form, abidnf 2947 is useful. (Contributed by NM, 8-Dec-2006.)
 |-  ( A  =  {
 z  |  A. x  z  e.  A }  ->  ( ph  <->  ps ) )   &    |-  ps   =>    |-  ( F/_ x A  ->  ph )
 
Theoremeqeu 2949* A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  ps  /\ 
 A. x ( ph  ->  x  =  A ) )  ->  E! x ph )
 
Theoremeueq 2950* Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
 |-  ( A  e.  _V  <->  E! x  x  =  A )
 
Theoremeueq1 2951* Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
 |-  A  e.  _V   =>    |-  E! x  x  =  A
 
Theoremeueq2 2952* Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 E! x ( (
 ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) )
 
Theoremeueq3 2953* Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  -.  ( ph  /\  ps )   =>    |-  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )
 
Theoremmoeq 2954* There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
 |- 
 E* x  x  =  A
 
Theoremmoeq3 2955* "At most one" property of equality (split into 3 cases). (The first 2 hypotheses could be eliminated with longer proof.) (Contributed by NM, 23-Apr-1995.)
 |-  B  e.  _V   &    |-  C  e.  _V   &    |-  -.  ( ph  /\ 
 ps )   =>    |- 
 E* x ( (
 ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C )
 )
 
Theoremmosub 2956* "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)
 |- 
 E* x ph   =>    |- 
 E* x E. y
 ( y  =  A  /\  ph )
 
Theoremmo2icl 2957* Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
 |-  ( A. x (
 ph  ->  x  =  A )  ->  E* x ph )
 
Theoremmob2 2958* Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  <->  ps ) )
 
Theoremmoi2 2959* Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ( ph  /\  ps )
 )  ->  x  =  A )
 
Theoremmob 2960* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( ( A  e.  C  /\  B  e.  D )  /\  E* x ph  /\  ps )  ->  ( A  =  B 
 <->  ch ) )
 
Theoremmoi 2961* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( ( A  e.  C  /\  B  e.  D )  /\  E* x ph  /\  ( ps  /\  ch ) ) 
 ->  A  =  B )
 
Theoremmorex 2962* Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  B  e.  _V   &    |-  ( x  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( E. x  e.  A  ph  /\  E* x ph )  ->  ( ps  ->  B  e.  A ) )
 
Theoremeuxfr2 2963* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 14-Nov-2004.)
 |-  A  e.  _V   &    |-  E* y  x  =  A   =>    |-  ( E! x E. y ( x  =  A  /\  ph )  <->  E! y ph )
 
Theoremeuxfr 2964* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 14-Nov-2004.)
 |-  A  e.  _V   &    |-  E! y  x  =  A   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E! x ph  <->  E! y ps )
 
Theoremeuind 2965* Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)
 |-  B  e.  _V   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  y  ->  A  =  B )   =>    |-  ( ( A. x A. y ( (
 ph  /\  ps )  ->  A  =  B ) 
 /\  E. x ph )  ->  E! z A. x ( ph  ->  z  =  A ) )
 
Theoremreu2 2966* A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)
 |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph  /\  A. x  e.  A  A. y  e.  A  ( ( ph  /\ 
 [ y  /  x ] ph )  ->  x  =  y ) ) )
 
Theoremreu6 2967* A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)
 |-  ( E! x  e.  A  ph  <->  E. y  e.  A  A. x  e.  A  (
 ph 
 <->  x  =  y ) )
 
Theoremreu3 2968* A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)
 |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph  /\  E. y  e.  A  A. x  e.  A  ( ph  ->  x  =  y ) ) )
 
Theoremreu6i 2969* A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( ( B  e.  A  /\  A. x  e.  A  ( ph  <->  x  =  B ) )  ->  E! x  e.  A  ph )
 
Theoremeqreu 2970* A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( ( B  e.  A  /\  ps  /\ 
 A. x  e.  A  ( ph  ->  x  =  B ) )  ->  E! x  e.  A  ph )
 
Theoremrmo4 2971* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E* x  e.  A ph  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremreu4 2972* Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph  /\  A. x  e.  A  A. y  e.  A  ( ( ph  /\ 
 ps )  ->  x  =  y ) ) )
 
Theoremreu7 2973* Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph  /\  E. x  e.  A  A. y  e.  A  ( ps  ->  x  =  y ) ) )
 
Theoremreu8 2974* Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  x  =  y ) ) )
 
Theoremreueq 2975* Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  ( B  e.  A  <->  E! x  e.  A  x  =  B )
 
Theoremrmoan 2976 Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)
 |-  ( E* x  e.  A ph  ->  E* x  e.  A ( ps  /\  ph ) )
 
Theoremrmoim 2977 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( A. x  e.  A  ( ph  ->  ps )  ->  ( E* x  e.  A ps  ->  E* x  e.  A ph ) )
 
Theoremrmoimia 2978 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( x  e.  A  ->  ( ph  ->  ps )
 )   =>    |-  ( E* x  e.  A ps  ->  E* x  e.  A ph )
 
Theoremrmoimi2 2979 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |- 
 A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ps )
 )   =>    |-  ( E* x  e.  B ps  ->  E* x  e.  A ph )
 
Theorem2reuswap 2980* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
 |-  ( A. x  e.  A  E* y  e.  B ph  ->  ( E! x  e.  A  E. y  e.  B  ph 
 ->  E! y  e.  B  E. x  e.  A  ph ) )
 
Theoremreuind 2981* Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  A  =  B )   =>    |-  ( ( A. x A. y ( ( ( A  e.  C  /\  ph )  /\  ( B  e.  C  /\  ps ) )  ->  A  =  B )  /\  E. x ( A  e.  C  /\  ph ) )  ->  E! z  e.  C  A. x ( ( A  e.  C  /\  ph )  ->  z  =  A ) )
 
Theorem2rmorex 2982* Double restricted quantification with "at most one," analogous to 2moex 2227. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( E* x  e.  A E. y  e.  B  ph  ->  A. y  e.  B  E* x  e.  A ph )
 
Theorem2reu5lem1 2983* Lemma for 2reu5 2986. Note that  E! x  e.  A E! y  e.  B ph does not mean "there is exactly one  x in  A and exactly one  y in  B such that  ph holds;" see comment for 2eu5 2240. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( E! x  e.  A  E! y  e.  B  ph  <->  E! x E! y
 ( x  e.  A  /\  y  e.  B  /\  ph ) )
 
Theorem2reu5lem2 2984* Lemma for 2reu5 2986. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( A. x  e.  A  E* y  e.  B ph  <->  A. x E* y
 ( x  e.  A  /\  y  e.  B  /\  ph ) )
 
Theorem2reu5lem3 2985* Lemma for 2reu5 2986. This lemma is interesting in its own right, showing that existential restriction in the last conjunct (the "at most one" part) is optional; compare rmo2 3089. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( ( E! x  e.  A  E! y  e.  B  ph  /\  A. x  e.  A  E* y  e.  B ph )  <->  ( E. x  e.  A  E. y  e.  B  ph  /\  E. z E. w A. x  e.  A  A. y  e.  B  ( ph  ->  ( x  =  z  /\  y  =  w )
 ) ) )
 
Theorem2reu5 2986* Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2240 and reu3 2968. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( ( E! x  e.  A  E! y  e.  B  ph  /\  A. x  e.  A  E* y  e.  B ph )  <->  ( E. x  e.  A  E. y  e.  B  ph  /\  E. z  e.  A  E. w  e.  B  A. x  e.  A  A. y  e.  B  ( ph  ->  ( x  =  z  /\  y  =  w )
 ) ) )
 
2.1.7  Conditional equality (experimental)

This is a very useless definition, which "abbreviates"  ( x  =  y  ->  ph ) as CondEq ( x  =  y  ->  ph ). What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific 3-ary operation  ( x  =  y  ->  ph ).

This is all used as part of a metatheorem: we want to say that  |-  ( x  =  y  ->  ( ph ( x )  <->  ph ( y ) ) ) and  |-  ( x  =  y  ->  A
( x )  =  A ( y ) ) are provable, for any expressions  ph ( x ) or  A ( x ) in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations.

The metatheorem comes with a disjoint variables assumption: every variable in  ph ( x ) is assumed disjoint from 
x except  x itself. For such a proof by induction, we must consider each of the possible forms of  ph ( x ). If it is a variable other than  x, then we have CondEq ( x  =  y  ->  A  =  A ) or CondEq ( x  =  y  ->  ( ph  <->  ph ) ), which is provable by cdeqth 2991 and reflexivity. Since we are only working with class and wff expressions, it can't be  x itself in set.mm, but if it was we'd have to also prove CondEq
( x  =  y  ->  x  =  y ) (where set equality is being used on the right).

Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each set variable parameter to the operation, we must consider if it is equal to  x or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one set variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder).

In each of the primitive proofs, we are allowed to assume that  y is disjoint from  ph ( x ) and vice versa, because this is maintained through the induction. This is how we satisfy the DV assumptions of cdeqab1 2996 and cdeqab 2994.

 
Syntaxwcdeq 2987 Extend wff notation to include conditional equality. This is a technical device used in the proof that 
F/ is the not-free predicate, and that definitions are conservative as a result.
 wff CondEq ( x  =  y 
 ->  ph )
 
Definitiondf-cdeq 2988 Define conditional equality. All the notation to the left of the  <-> is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq x y ph. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  (CondEq ( x  =  y  ->  ph )  <->  ( x  =  y  ->  ph ) )
 
Theoremcdeqi 2989 Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( x  =  y 
 ->  ph )   =>    |- CondEq ( x  =  y  -> 
 ph )
 
Theoremcdeqri 2990 Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  -> 
 ph )   =>    |-  ( x  =  y 
 ->  ph )
 
Theoremcdeqth 2991 Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ph   =>    |- CondEq ( x  =  y  -> 
 ph )
 
Theoremcdeqnot 2992 Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( -.  ph  <->  -. 
 ps ) )
 
Theoremcdeqal 2993* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( A. z ph  <->  A. z ps )
 )
 
Theoremcdeqab 2994* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- CondEq ( x  =  y  ->  { z  |  ph }  =  {
 z  |  ps }
 )
 
Theoremcdeqal1 2995* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( A. x ph  <->  A. y ps )
 )
 
Theoremcdeqab1 2996* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- CondEq ( x  =  y  ->  { x  |  ph }  =  {
 y  |  ps }
 )
 
Theoremcdeqim 2997 Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   &    |- CondEq ( x  =  y  ->  ( ch 
 <-> 
 th ) )   =>    |- CondEq ( x  =  y  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) )
 
Theoremcdeqcv 2998 Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  x  =  y )
 
Theoremcdeqeq 2999 Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  A  =  B )   &    |- CondEq ( x  =  y  ->  C  =  D )   =>    |- CondEq ( x  =  y  ->  ( A  =  C  <->  B  =  D ) )
 
Theoremcdeqel 3000 Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  A  =  B )   &    |- CondEq ( x  =  y  ->  C  =  D )   =>    |- CondEq ( x  =  y  ->  ( A  e.  C  <->  B  e.  D ) )
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