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Theorem List for Metamath Proof Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremr19.26m 2801 Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by NM, 22-Feb-2004.)
 |-  ( A. x ( ( x  e.  A  -> 
 ph )  /\  ( x  e.  B  ->  ps ) )  <->  ( A. x  e.  A  ph  /\  A. x  e.  B  ps ) )
 
Theoremralbi 2802 Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification. (Contributed by NM, 6-Oct-2003.)
 |-  ( A. x  e.  A  ( ph  <->  ps )  ->  ( A. x  e.  A  ph  <->  A. x  e.  A  ps ) )
 
Theoremralbiim 2803 Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)
 |-  ( A. x  e.  A  ( ph  <->  ps )  <->  ( A. x  e.  A  ( ph  ->  ps )  /\  A. x  e.  A  ( ps  ->  ph ) ) )
 
Theoremr19.27av 2804* Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when  A is empty.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |-  ( ( A. x  e.  A  ph  /\  ps )  ->  A. x  e.  A  ( ph  /\  ps )
 )
 
Theoremr19.28av 2805* Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when  A is empty.) (Contributed by NM, 2-Apr-2004.)
 |-  ( ( ph  /\  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  /\  ps )
 )
 
Theoremr19.29 2806 Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |-  ( ( A. x  e.  A  ph  /\  E. x  e.  A  ps )  ->  E. x  e.  A  ( ph  /\  ps )
 )
 
Theoremr19.29r 2807 Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
 |-  ( ( E. x  e.  A  ph  /\  A. x  e.  A  ps )  ->  E. x  e.  A  ( ph  /\  ps )
 )
 
Theoremr19.29af2 2808 A commonly used pattern based on r19.29 2806 (Contributed by Thierry Arnoux, 17-Dec-2017.)
 |- 
 F/ x ph   &    |-  F/ x ch   &    |-  (
 ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  ps )   =>    |-  ( ph  ->  ch )
 
Theoremr19.29af 2809* A commonly used pattern based on r19.29 2806 (Contributed by Thierry Arnoux, 29-Nov-2017.)
 |- 
 F/ x ph   &    |-  ( ( (
 ph  /\  x  e.  A )  /\  ps )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  ps )   =>    |-  ( ph  ->  ch )
 
Theoremr19.29a 2810* A commonly used pattern based on r19.29 2806 (Contributed by Thierry Arnoux, 22-Nov-2017.)
 |-  ( ( ( ph  /\  x  e.  A ) 
 /\  ps )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  ps )   =>    |-  ( ph  ->  ch )
 
Theoremr19.29d2r 2811 Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )   &    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ch )   =>    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ( ps  /\  ch ) )
 
Theoremr19.29_2a 2812* A commonly used pattern based on r19.29 2806, version with two restricted quantifiers (Contributed by Thierry Arnoux, 26-Nov-2017.)
 |-  ( ( ( (
 ph  /\  x  e.  A )  /\  y  e.  B )  /\  ps )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )   =>    |-  ( ph  ->  ch )
 
Theoremr19.30 2813 Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  ( A. x  e.  A  ( ph  \/  ps )  ->  ( A. x  e.  A  ph  \/  E. x  e.  A  ps ) )
 
Theoremr19.32v 2814* Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 25-Nov-2003.)
 |-  ( A. x  e.  A  ( ph  \/  ps )  <->  ( ph  \/  A. x  e.  A  ps ) )
 
Theoremr19.35 2815 Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 20-Sep-2003.)
 |-  ( E. x  e.  A  ( ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )
 
Theoremr19.36av 2816* One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. The other direction doesn't hold when  A is empty. (Contributed by NM, 22-Oct-2003.)
 |-  ( E. x  e.  A  ( ph  ->  ps )  ->  ( A. x  e.  A  ph  ->  ps ) )
 
Theoremr19.37 2817 Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when  A is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |- 
 F/ x ph   =>    |-  ( E. x  e.  A  ( ph  ->  ps )  ->  ( ph  ->  E. x  e.  A  ps ) )
 
Theoremr19.37av 2818* Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when  A is empty.) (Contributed by NM, 2-Apr-2004.)
 |-  ( E. x  e.  A  ( ph  ->  ps )  ->  ( ph  ->  E. x  e.  A  ps ) )
 
Theoremr19.40 2819 Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
 |-  ( E. x  e.  A  ( ph  /\  ps )  ->  ( E. x  e.  A  ph  /\  E. x  e.  A  ps ) )
 
Theoremr19.41 2820 Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
 |- 
 F/ x ps   =>    |-  ( E. x  e.  A  ( ph  /\  ps ) 
 <->  ( E. x  e.  A  ph  /\  ps )
 )
 
Theoremr19.41v 2821* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)
 |-  ( E. x  e.  A  ( ph  /\  ps ) 
 <->  ( E. x  e.  A  ph  /\  ps )
 )
 
Theoremr19.42v 2822* Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
 |-  ( E. x  e.  A  ( ph  /\  ps ) 
 <->  ( ph  /\  E. x  e.  A  ps ) )
 
Theoremr19.43 2823 Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |-  ( E. x  e.  A  ( ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
 
Theoremr19.44av 2824* One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when  A is empty. (Contributed by NM, 2-Apr-2004.)
 |-  ( E. x  e.  A  ( ph  \/  ps )  ->  ( E. x  e.  A  ph  \/  ps ) )
 
Theoremr19.45av 2825* Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when  A is empty.) (Contributed by NM, 2-Apr-2004.)
 |-  ( E. x  e.  A  ( ph  \/  ps )  ->  ( ph  \/  E. x  e.  A  ps ) )
 
Theoremralcomf 2826* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ y A   &    |-  F/_ x B   =>    |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. y  e.  B  A. x  e.  A  ph )
 
Theoremrexcomf 2827* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ y A   &    |-  F/_ x B   =>    |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. y  e.  B  E. x  e.  A  ph )
 
Theoremralcom 2828* Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. y  e.  B  A. x  e.  A  ph )
 
Theoremrexcom 2829* Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. y  e.  B  E. x  e.  A  ph )
 
Theoremrexcom13 2830* Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
 |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  ph  <->  E. z  e.  C  E. y  e.  B  E. x  e.  A  ph )
 
Theoremrexrot4 2831* Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
 |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  E. w  e.  D  ph  <->  E. z  e.  C  E. w  e.  D  E. x  e.  A  E. y  e.  B  ph )
 
Theoremralcom2 2832* Commutation of restricted quantifiers. Note that  x and  y needn't be distinct (this makes the proof longer). (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)
 |-  ( A. x  e.  A  A. y  e.  A  ph  ->  A. y  e.  A  A. x  e.  A  ph )
 
Theoremralcom3 2833 A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)
 |-  ( A. x  e.  A  ( x  e.  B  ->  ph )  <->  A. x  e.  B  ( x  e.  A  -> 
 ph ) )
 
Theoremreean 2834* Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps )  <->  ( E. x  e.  A  ph 
 /\  E. y  e.  B  ps ) )
 
Theoremreeanv 2835* Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
 |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps ) 
 <->  ( E. x  e.  A  ph  /\  E. y  e.  B  ps ) )
 
Theorem3reeanv 2836* Rearrange three existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
 |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  ( ph  /\  ps  /\ 
 ch )  <->  ( E. x  e.  A  ph  /\  E. y  e.  B  ps  /\  E. z  e.  C  ch ) )
 
Theorem2ralor 2837* Distribute quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  ( A. x  e.  A  A. y  e.  B  ( ph  \/  ps )  <->  ( A. x  e.  A  ph  \/  A. y  e.  B  ps ) )
 
Theoremnfreu1 2838  x is not free in  E! x  e.  A ph. (Contributed by NM, 19-Mar-1997.)
 |- 
 F/ x E! x  e.  A  ph
 
Theoremnfrmo1 2839  x is not free in  E* x  e.  A ph. (Contributed by NM, 16-Jun-2017.)
 |- 
 F/ x E* x  e.  A ph
 
Theoremnfreud 2840 Deduction version of nfreu 2842. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E! y  e.  A  ps )
 
Theoremnfrmod 2841 Deduction version of nfrmo 2843. (Contributed by NM, 17-Jun-2017.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E* y  e.  A ps )
 
Theoremnfreu 2842 Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x E! y  e.  A  ph
 
Theoremnfrmo 2843 Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 16-Jun-2017.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x E* y  e.  A ph
 
Theoremrabid 2844 An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)
 |-  ( x  e.  { x  e.  A  |  ph
 } 
 <->  ( x  e.  A  /\  ph ) )
 
Theoremrabid2 2845* An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |-  ( A  =  { x  e.  A  |  ph
 } 
 <-> 
 A. x  e.  A  ph )
 
Theoremrabbi 2846 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2907. (Contributed by NM, 25-Nov-2013.)
 |-  ( A. x  e.  A  ( ps  <->  ch )  <->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch } )
 
Theoremrabswap 2847 Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
 |- 
 { x  e.  A  |  x  e.  B }  =  { x  e.  B  |  x  e.  A }
 
Theoremnfrab1 2848 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
 |-  F/_ x { x  e.  A  |  ph }
 
Theoremnfrab 2849 A variable not free in a wff remains so in a restricted class abstraction. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.)
 |- 
 F/ x ph   &    |-  F/_ x A   =>    |-  F/_ x { y  e.  A  |  ph }
 
Theoremreubida 2850 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)
 |- 
 F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E! x  e.  A  ps 
 <->  E! x  e.  A  ch ) )
 
Theoremreubidva 2851* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 13-Nov-2004.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E! x  e.  A  ps 
 <->  E! x  e.  A  ch ) )
 
Theoremreubidv 2852* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E! x  e.  A  ps 
 <->  E! x  e.  A  ch ) )
 
Theoremreubiia 2853 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
 |-  ( x  e.  A  ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  E! x  e.  A  ps )
 
Theoremreubii 2854 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.)
 |-  ( ph  <->  ps )   =>    |-  ( E! x  e.  A  ph  <->  E! x  e.  A  ps )
 
Theoremrmobida 2855 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
 |- 
 F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E* x  e.  A ps 
 <->  E* x  e.  A ch ) )
 
Theoremrmobidva 2856* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E* x  e.  A ps 
 <->  E* x  e.  A ch ) )
 
Theoremrmobidv 2857* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E* x  e.  A ps 
 <->  E* x  e.  A ch ) )
 
Theoremrmobiia 2858 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
 |-  ( x  e.  A  ->  ( ph  <->  ps ) )   =>    |-  ( E* x  e.  A ph  <->  E* x  e.  A ps )
 
Theoremrmobii 2859 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
 |-  ( ph  <->  ps )   =>    |-  ( E* x  e.  A ph  <->  E* x  e.  A ps )
 
Theoremraleqf 2860 Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )
 
Theoremrexeqf 2861 Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
 
Theoremreueq1f 2862 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
 
Theoremrmoeq1f 2863 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  =  B  ->  ( E* x  e.  A ph  <->  E* x  e.  B ph ) )
 
Theoremraleq 2864* Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
 |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )
 
Theoremrexeq 2865* Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
 |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
 
Theoremreueq1 2866* Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
 |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
 
Theoremrmoeq1 2867* Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( A  =  B  ->  ( E* x  e.  A ph  <->  E* x  e.  B ph ) )
 
Theoremraleqi 2868* Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  A  =  B   =>    |-  ( A. x  e.  A  ph  <->  A. x  e.  B  ph )
 
Theoremrexeqi 2869* Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  A  =  B   =>    |-  ( E. x  e.  A  ph  <->  E. x  e.  B  ph )
 
Theoremraleqdv 2870* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A. x  e.  A  ps 
 <-> 
 A. x  e.  B  ps ) )
 
Theoremrexeqdv 2871* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  B  ps ) )
 
Theoremraleqbi1dv 2872* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
 |-  ( A  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ps ) )
 
Theoremrexeqbi1dv 2873* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
 |-  ( A  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ps ) )
 
Theoremreueqd 2874* Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
 |-  ( A  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ps ) )
 
Theoremrmoeqd 2875* Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( A  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( A  =  B  ->  ( E* x  e.  A ph  <->  E* x  e.  B ps ) )
 
Theoremraleqbidv 2876* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  B  ch ) )
 
Theoremrexeqbidv 2877* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  B  ch ) )
 
Theoremraleqbidva 2878* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  B  ch ) )
 
Theoremrexeqbidva 2879* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  B  ch ) )
 
Theoremmormo 2880 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
 |-  ( E* x ph  ->  E* x  e.  A ph )
 
Theoremreu5 2881 Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
 |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph  /\  E* x  e.  A ph ) )
 
Theoremreurex 2882 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
 |-  ( E! x  e.  A  ph  ->  E. x  e.  A  ph )
 
Theoremreurmo 2883 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
 |-  ( E! x  e.  A  ph  ->  E* x  e.  A ph )
 
Theoremrmo5 2884 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
 |-  ( E* x  e.  A ph  <->  ( E. x  e.  A  ph  ->  E! x  e.  A  ph ) )
 
Theoremnrexrmo 2885 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
 |-  ( -.  E. x  e.  A  ph  ->  E* x  e.  A ph )
 
Theoremcbvralf 2886 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )
 
Theoremcbvrexf 2887 Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
 
Theoremcbvral 2888* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )
 
Theoremcbvrex 2889* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
 
Theoremcbvreu 2890* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
 
Theoremcbvrmo 2891* Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x  e.  A ph  <->  E* y  e.  A ps )
 
Theoremcbvralv 2892* Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )
 
Theoremcbvrexv 2893* Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
 
Theoremcbvreuv 2894* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
 
Theoremcbvrmov 2895* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E* x  e.  A ph  <->  E* y  e.  A ps )
 
Theoremcbvraldva2 2896* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   &    |-  (
 ( ph  /\  x  =  y )  ->  A  =  B )   =>    |-  ( ph  ->  ( A. x  e.  A  ps 
 <-> 
 A. y  e.  B  ch ) )
 
Theoremcbvrexdva2 2897* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   &    |-  (
 ( ph  /\  x  =  y )  ->  A  =  B )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. y  e.  B  ch ) )
 
Theoremcbvraldva 2898* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. y  e.  A  ch ) )
 
Theoremcbvrexdva 2899* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. y  e.  A  ch ) )
 
Theoremcbvral2v 2900* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
 |-  ( x  =  z 
 ->  ( ph  <->  ch ) )   &    |-  (
 y  =  w  ->  ( ch  <->  ps ) )   =>    |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. z  e.  A  A. w  e.  B  ps )
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