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Theorem List for Metamath Proof Explorer - 2601-2700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrgen2 2601* Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)
 |-  ( ( x  e.  A  /\  y  e.  B )  ->  ph )   =>    |-  A. x  e.  A  A. y  e.  B  ph
 
Theoremrgen3 2602* Generalization rule for restricted quantification. (Contributed by NM, 12-Jan-2008.)
 |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  ->  ph )   =>    |-  A. x  e.  A  A. y  e.  B  A. z  e.  C  ph
 
Theoremr19.21bi 2603 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
 |-  ( ph  ->  A. x  e.  A  ps )   =>    |-  ( ( ph  /\  x  e.  A ) 
 ->  ps )
 
Theoremrspec2 2604 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
 |- 
 A. x  e.  A  A. y  e.  B  ph   =>    |-  (
 ( x  e.  A  /\  y  e.  B )  ->  ph )
 
Theoremrspec3 2605 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
 |- 
 A. x  e.  A  A. y  e.  B  A. z  e.  C  ph   =>    |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  ->  ph )
 
Theoremr19.21be 2606 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)
 |-  ( ph  ->  A. x  e.  A  ps )   =>    |-  A. x  e.  A  ( ph  ->  ps )
 
Theoremnrex 2607 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
 |-  ( x  e.  A  ->  -.  ps )   =>    |-  -.  E. x  e.  A  ps
 
Theoremnrexdv 2608* Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
 |-  ( ( ph  /\  x  e.  A )  ->  -.  ps )   =>    |-  ( ph  ->  -.  E. x  e.  A  ps )
 
Theoremrexim 2609 Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |-  ( A. x  e.  A  ( ph  ->  ps )  ->  ( E. x  e.  A  ph  ->  E. x  e.  A  ps ) )
 
Theoremreximia 2610 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
 |-  ( x  e.  A  ->  ( ph  ->  ps )
 )   =>    |-  ( E. x  e.  A  ph  ->  E. x  e.  A  ps )
 
Theoremreximi2 2611 Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.)
 |-  ( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ps ) )   =>    |-  ( E. x  e.  A  ph  ->  E. x  e.  B  ps )
 
Theoremreximi 2612 Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x  e.  A  ph 
 ->  E. x  e.  A  ps )
 
Theoremreximdai 2613 Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch )
 ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ch ) )
 
Theoremreximdv2 2614* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)
 |-  ( ph  ->  (
 ( x  e.  A  /\  ps )  ->  ( x  e.  B  /\  ch ) ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  E. x  e.  B  ch ) )
 
Theoremreximdvai 2615* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.)
 |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch )
 ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ch ) )
 
Theoremreximdv 2616* Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ch ) )
 
Theoremreximdva 2617* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ch ) )
 
Theoremr19.12 2618* Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |-  ( E. x  e.  A  A. y  e.  B  ph  ->  A. y  e.  B  E. x  e.  A  ph )
 
Theoremr19.23t 2619 Closed theorem form of r19.23 2620. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
 |-  ( F/ x ps  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph  ->  ps )
 ) )
 
Theoremr19.23 2620 Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)
 |- 
 F/ x ps   =>    |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph  ->  ps )
 )
 
Theoremr19.23v 2621* Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
 |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph  ->  ps )
 )
 
Theoremrexlimi 2622 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |- 
 F/ x ps   &    |-  ( x  e.  A  ->  (
 ph  ->  ps ) )   =>    |-  ( E. x  e.  A  ph  ->  ps )
 
Theoremrexlimiv 2623* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
 |-  ( x  e.  A  ->  ( ph  ->  ps )
 )   =>    |-  ( E. x  e.  A  ph  ->  ps )
 
Theoremrexlimiva 2624* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)
 |-  ( ( x  e.  A  /\  ph )  ->  ps )   =>    |-  ( E. x  e.  A  ph  ->  ps )
 
Theoremrexlimivw 2625* Weaker version of rexlimiv 2623. (Contributed by FL, 19-Sep-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x  e.  A  ph 
 ->  ps )
 
Theoremrexlimd 2626 Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |- 
 F/ x ph   &    |-  F/ x ch   &    |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
 
Theoremrexlimd2 2627 Version of rexlimd 2626 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
 
Theoremrexlimdv 2628* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 14-Nov-2002.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
 |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch )
 ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
 
Theoremrexlimdva 2629* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
 
Theoremrexlimdvaa 2630* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)
 |-  ( ( ph  /\  ( x  e.  A  /\  ps ) )  ->  ch )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch )
 )
 
Theoremrexlimdv3a 2631* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2628. (Contributed by NM, 7-Jun-2015.)
 |-  ( ( ph  /\  x  e.  A  /\  ps )  ->  ch )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
 
Theoremrexlimdvw 2632* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
 
Theoremrexlimddv 2633* Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
 |-  ( ph  ->  E. x  e.  A  ps )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  ps ) )  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremrexlimivv 2634* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)
 |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( ph  ->  ps ) )   =>    |-  ( E. x  e.  A  E. y  e.  B  ph  ->  ps )
 
Theoremrexlimdvv 2635* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)
 |-  ( ph  ->  (
 ( x  e.  A  /\  y  e.  B )  ->  ( ps  ->  ch ) ) )   =>    |-  ( ph  ->  ( E. x  e.  A  E. y  e.  B  ps  ->  ch ) )
 
Theoremrexlimdvva 2636* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  E. y  e.  B  ps  ->  ch ) )
 
Theoremr19.26 2637 Theorem 19.26 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |-  ( A. x  e.  A  ( ph  /\  ps ) 
 <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
 
Theoremr19.26-2 2638 Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers. (Contributed by NM, 10-Aug-2004.)
 |-  ( A. x  e.  A  A. y  e.  B  ( ph  /\  ps ) 
 <->  ( A. x  e.  A  A. y  e.  B  ph  /\  A. x  e.  A  A. y  e.  B  ps ) )
 
Theoremr19.26-3 2639 Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 22-Nov-2010.)
 |-  ( A. x  e.  A  ( ph  /\  ps  /\ 
 ch )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps  /\  A. x  e.  A  ch ) )
 
Theoremr19.26m 2640 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 2641 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 2642 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 2643* 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 2644* 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 2645 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 2646 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 )
 )
 
Theoremr19.30 2647 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 2648* 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 2649 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 2650* 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 2651 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 2652* 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 2653 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 2654 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 2655* 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 2656* 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 2657 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 2658* 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 2659* 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 2660* 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 2661* 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 2662* 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 2663* 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 2664* 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 2665* 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 2666* 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 2667 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 2668* 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 2669* 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 2670* 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 2671* 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 2672  x is not free in  E! x  e.  A ph. (Contributed by NM, 19-Mar-1997.)
 |- 
 F/ x E! x  e.  A  ph
 
Theoremnfreud 2673 Deduction version of nfreu 2674. (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 )
 
Theoremnfreu 2674 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
 
Theoremrabid 2675 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 2676* 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 2677 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2718. (Contributed by NM, 25-Nov-2013.)
 |-  ( A. x  e.  A  ( ps  <->  ch )  <->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch } )
 
Theoremrabswap 2678 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 2679 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
 |-  F/_ x { x  e.  A  |  ph }
 
Theoremnfrab 2680 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 2681 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 2682* 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 2683* 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 2684 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 2685 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 )
 
Theoremraleqf 2686 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 2687 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 2688 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 ) )
 
Theoremraleq 2689* 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 2690* 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 2691* Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
 |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
 
Theoremraleqi 2692* 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 2693* 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 2694* 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 2695* 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 2696* 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 2697* 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 2698* 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 ) )
 
Theoremraleqbidv 2699* 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 2700* 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 ) )
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