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Theorem List for Metamath Proof Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremspc3gv 3001* Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
 |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A. x A. y A. z ph  ->  ps ) )
 
Theoremspcv 3002* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremspcev 3003* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ps  ->  E. x ph )
 
Theoremspc2ev 3004* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ps  ->  E. x E. y ph )
 
Theoremrspct 3005* A closed version of rspc 3006. (Contributed by Andrew Salmon, 6-Jun-2011.)
 |- 
 F/ x ps   =>    |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps )
 ) )
 
Theoremrspc 3006* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps )
 )
 
Theoremrspce 3007* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )
 
Theoremrspcv 3008* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps )
 )
 
Theoremrspccv 3009* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  B  ph  ->  ( A  e.  B  ->  ps )
 )
 
Theoremrspcva 3010* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  A. x  e.  B  ph )  ->  ps )
 
Theoremrspccva 3011* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A. x  e.  B  ph  /\  A  e.  B )  ->  ps )
 
Theoremrspcev 3012* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )
 
Theoremrspcimdv 3013* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
 
Theoremrspcimedv 3014* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ch  ->  ps )
 )   =>    |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps ) )
 
Theoremrspcdv 3015* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  B  ps  ->  ch ) )
 
Theoremrspcedv 3016* 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 3017* 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 3018* 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 3019* 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 3020* 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 3021* 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 3022* 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 3023* 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 3024 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 3025* 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 3026* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
 |-  ( x  =  A  ->  ( ph  <->  E. x ( x  =  A  /\  ph )
 ) )
 
Theoremceqsexg 3027* 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 3028* 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 3029* 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 3030* 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 3031* 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 3032* 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 3033* 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 3034* 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 3035* 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 3036* 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 3037* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the non-empty class condition of r19.3rzv 3681 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 3038* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the non-empty class condition of r19.28zv 3683 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 3039* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3043.) (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 3040 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 3041* 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 3042* 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 3043* 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 3044* 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 3045* 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 3046* 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 3047 Membership in a class abstraction, with a weaker antecedent than elabgf 3040. (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 3048* Membership in a class abstraction, with a weaker antecedent than elabg 3043. (Contributed by NM, 29-Aug-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( ps 
 ->  A  e.  B ) 
 ->  ( A  e.  { x  |  ph }  <->  ps ) )
 
Theoremelab3 3049* 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 )
 
Theoremelrabi 3050* Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
 |-  ( A  e.  { x  e.  V  |  ph
 }  ->  A  e.  V )
 
Theoremelrabf 3051 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 3052* 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 3053* 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 3054* 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 3055* 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 3056* 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 3057* 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 3058* 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 3059* 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 3060* 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 3061* 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 3062* 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 3063* 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 3064* A deduction theorem for converting the inference  |- 
F/_ x A =>  |-  ph into a closed theorem. Use nfa1 1802 and nfab 2544 to eliminate the hypothesis of the substitution instance  ps of the inference. For converting the inference form into a deduction form, abidnf 3063 is useful. (Contributed by NM, 8-Dec-2006.)
 |-  ( A  =  {
 z  |  A. x  z  e.  A }  ->  ( ph  <->  ps ) )   &    |-  ps   =>    |-  ( F/_ x A  ->  ph )
 
Theoremeqeu 3065* 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 3066* Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
 |-  ( A  e.  _V  <->  E! x  x  =  A )
 
Theoremeueq1 3067* Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
 |-  A  e.  _V   =>    |-  E! x  x  =  A
 
Theoremeueq2 3068* 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 3069* 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 3070* There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
 |- 
 E* x  x  =  A
 
Theoremmoeq3 3071* "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 3072* "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)
 |- 
 E* x ph   =>    |- 
 E* x E. y
 ( y  =  A  /\  ph )
 
Theoremmo2icl 3073* Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
 |-  ( A. x (
 ph  ->  x  =  A )  ->  E* x ph )
 
Theoremmob2 3074* 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 3075* 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 3076* 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 3077* 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 3078* 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 3079* 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 3080* 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 3081* 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 3082* 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 3083* 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 3084* 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 3085* 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 3086* 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 3087* 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 3088* 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 3089* 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 3090* 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 3091* Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  ( B  e.  A  <->  E! x  e.  A  x  =  B )
 
Theoremrmoan 3092 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 3093 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 3094 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 3095 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 3096* 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 3097* 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 3098* Double restricted quantification with "at most one," analogous to 2moex 2325. (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 3099* Lemma for 2reu5 3102. 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 2338. (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 3100* Lemma for 2reu5 3102. (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 ) )
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