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Theorem List for Metamath Proof Explorer - 2601-2700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremralinexa 2601 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
 |-  ( A. x  e.  A  ( ph  ->  -. 
 ps )  <->  -.  E. x  e.  A  ( ph  /\  ps ) )
 
Theoremrexanali 2602 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
 |-  ( E. x  e.  A  ( ph  /\  -.  ps )  <->  -.  A. x  e.  A  ( ph  ->  ps ) )
 
Theoremrisset 2603* Two ways to say " A belongs to  B." (Contributed by NM, 22-Nov-1994.)
 |-  ( A  e.  B  <->  E. x  e.  B  x  =  A )
 
Theoremhbral 2604 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)
 |-  ( y  e.  A  ->  A. x  y  e.  A )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( A. y  e.  A  ph  ->  A. x A. y  e.  A  ph )
 
Theoremhbra1 2605  x is not free in  A. x  e.  A ph. (Contributed by NM, 18-Oct-1996.)
 |-  ( A. x  e.  A  ph  ->  A. x A. x  e.  A  ph )
 
Theoremnfra1 2606  x is not free in  A. x  e.  A ph. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ x A. x  e.  A  ph
 
Theoremnfrald 2607 Deduction version of nfral 2609. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y  e.  A  ps )
 
Theoremnfrexd 2608 Deduction version of nfrex 2611. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y  e.  A  ps )
 
Theoremnfral 2609 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x A. y  e.  A  ph
 
Theoremnfra2 2610* Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 28952. Contributed by Alan Sare 31-Dec-2011. (Contributed by NM, 31-Dec-2011.)
 |- 
 F/ y A. x  e.  A  A. y  e.  B  ph
 
Theoremnfrex 2611 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x E. y  e.  A  ph
 
Theoremnfre1 2612  x is not free in  E. x  e.  A ph. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ x E. x  e.  A  ph
 
Theoremr3al 2613* Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
 |-  ( A. x  e.  A  A. y  e.  B  A. z  e.  C  ph  <->  A. x A. y A. z ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  ->  ph )
 )
 
Theoremalral 2614 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
 |-  ( A. x ph  ->  A. x  e.  A  ph )
 
Theoremrexex 2615 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
 |-  ( E. x  e.  A  ph  ->  E. x ph )
 
Theoremrsp 2616 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
 |-  ( A. x  e.  A  ph  ->  ( x  e.  A  ->  ph )
 )
 
Theoremrspe 2617 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
 |-  ( ( x  e.  A  /\  ph )  ->  E. x  e.  A  ph )
 
Theoremrsp2 2618 Restricted specialization. (Contributed by NM, 11-Feb-1997.)
 |-  ( A. x  e.  A  A. y  e.  B  ph  ->  ( ( x  e.  A  /\  y  e.  B )  -> 
 ph ) )
 
Theoremrsp2e 2619 Restricted specialization. (Contributed by FL, 4-Jun-2012.)
 |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  E. x  e.  A  E. y  e.  B  ph )
 
Theoremrspec 2620 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
 |- 
 A. x  e.  A  ph   =>    |-  ( x  e.  A  -> 
 ph )
 
Theoremrgen 2621 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
 |-  ( x  e.  A  -> 
 ph )   =>    |- 
 A. x  e.  A  ph
 
Theoremrgen2a 2622* Generalization rule for restricted quantification. Note that  x and  y needn't be distinct (and illustrates the use of dvelim 1969). (Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.
 |-  ( ( x  e.  A  /\  y  e.  A )  ->  ph )   =>    |-  A. x  e.  A  A. y  e.  A  ph
 
Theoremrgenw 2623 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
 |-  ph   =>    |- 
 A. x  e.  A  ph
 
Theoremrgen2w 2624 Generalization rule for restricted quantification. Note that  x and  y needn't be distinct. (Contributed by NM, 18-Jun-2014.)
 |-  ph   =>    |- 
 A. x  e.  A  A. y  e.  B  ph
 
Theoremmprg 2625 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
 |-  ( A. x  e.  A  ph  ->  ps )   &    |-  ( x  e.  A  ->  ph )   =>    |- 
 ps
 
Theoremmprgbir 2626 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
 |-  ( ph  <->  A. x  e.  A  ps )   &    |-  ( x  e.  A  ->  ps )   =>    |-  ph
 
Theoremralim 2627 Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
 |-  ( A. x  e.  A  ( ph  ->  ps )  ->  ( A. x  e.  A  ph  ->  A. x  e.  A  ps ) )
 
Theoremralimi2 2628 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
 |-  ( ( x  e.  A  ->  ph )  ->  ( x  e.  B  ->  ps ) )   =>    |-  ( A. x  e.  A  ph  ->  A. x  e.  B  ps )
 
Theoremralimia 2629 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
 |-  ( x  e.  A  ->  ( ph  ->  ps )
 )   =>    |-  ( A. x  e.  A  ph  ->  A. x  e.  A  ps )
 
Theoremralimiaa 2630 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
 |-  ( ( x  e.  A  /\  ph )  ->  ps )   =>    |-  ( A. x  e.  A  ph  ->  A. x  e.  A  ps )
 
Theoremralimi 2631 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x  e.  A  ph 
 ->  A. x  e.  A  ps )
 
Theoremral2imi 2632 Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x  e.  A  ph  ->  ( A. x  e.  A  ps  ->  A. x  e.  A  ch ) )
 
Theoremralimdaa 2633 Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)
 |- 
 F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x  e.  A  ps  ->  A. x  e.  A  ch ) )
 
Theoremralimdva 2634* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps  ->  A. x  e.  A  ch ) )
 
Theoremralimdv 2635* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps  ->  A. x  e.  A  ch ) )
 
Theoremralimdv2 2636* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
 |-  ( ph  ->  (
 ( x  e.  A  ->  ps )  ->  ( x  e.  B  ->  ch ) ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps  ->  A. x  e.  B  ch ) )
 
Theoremralrimi 2637 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( x  e.  A  ->  ps ) )   =>    |-  ( ph  ->  A. x  e.  A  ps )
 
Theoremralrimiv 2638* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ph  ->  ( x  e.  A  ->  ps ) )   =>    |-  ( ph  ->  A. x  e.  A  ps )
 
Theoremralrimiva 2639* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)
 |-  ( ( ph  /\  x  e.  A )  ->  ps )   =>    |-  ( ph  ->  A. x  e.  A  ps )
 
Theoremralrimivw 2640* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x  e.  A  ps )
 
Theoremr19.21t 2641 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)
 |-  ( F/ x ph  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  ( ph  ->  A. x  e.  A  ps ) ) )
 
Theoremr19.21 2642 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)
 |- 
 F/ x ph   =>    |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( ph  ->  A. x  e.  A  ps ) )
 
Theoremr19.21v 2643* Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( ph  ->  A. x  e.  A  ps ) )
 
Theoremralrimd 2644 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)
 |- 
 F/ x ph   &    |-  F/ x ps   &    |-  ( ph  ->  ( ps  ->  ( x  e.  A  ->  ch ) ) )   =>    |-  ( ph  ->  ( ps  ->  A. x  e.  A  ch ) )
 
Theoremralrimdv 2645* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)
 |-  ( ph  ->  ( ps  ->  ( x  e.  A  ->  ch )
 ) )   =>    |-  ( ph  ->  ( ps  ->  A. x  e.  A  ch ) )
 
Theoremralrimdva 2646* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x  e.  A  ch ) )
 
Theoremralrimivv 2647* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)
 |-  ( ph  ->  (
 ( x  e.  A  /\  y  e.  B )  ->  ps ) )   =>    |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )
 
Theoremralrimivva 2648* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ps )   =>    |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )
 
Theoremralrimivvva 2649* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B  /\  z  e.  C )
 )  ->  ps )   =>    |-  ( ph  ->  A. x  e.  A  A. y  e.  B  A. z  e.  C  ps )
 
Theoremralrimdvv 2650* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)
 |-  ( ph  ->  ( ps  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ch )
 ) )   =>    |-  ( ph  ->  ( ps  ->  A. x  e.  A  A. y  e.  B  ch ) )
 
Theoremralrimdvva 2651* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x  e.  A  A. y  e.  B  ch ) )
 
Theoremrgen2 2652* 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 2653* 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 2654 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 2655 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 2656 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 2657 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 2658 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
 |-  ( x  e.  A  ->  -.  ps )   =>    |-  -.  E. x  e.  A  ps
 
Theoremnrexdv 2659* 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 2660 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 2661 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 2662 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 2663 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 2664 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 2665* 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 2666* 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 2667* 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 2668* 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 2669* 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 2670 Closed theorem form of r19.23 2671. (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 2671 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 2672* 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 2673 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 2674* 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 2675* 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 2676* Weaker version of rexlimiv 2674. (Contributed by FL, 19-Sep-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x  e.  A  ph 
 ->  ps )
 
Theoremrexlimd 2677 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 2678 Version of rexlimd 2677 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 2679* 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 2680* 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 2681* 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 2682* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2679. (Contributed by NM, 7-Jun-2015.)
 |-  ( ( ph  /\  x  e.  A  /\  ps )  ->  ch )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
 
Theoremrexlimdvw 2683* 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 2684* 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 2685* 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 2686* 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 2687* 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 2688 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 2689 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 2690 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 2691 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 2692 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 2693 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 2694* 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 2695* 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 2696 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 2697 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 2698 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 2699* 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 2700 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 ) )
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