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Theorem List for Intuitionistic Logic Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremralinexa 2401 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 ) )
 
Theoremrisset 2402* Two ways to say " A belongs to  B." (Contributed by NM, 22-Nov-1994.)
 |-  ( A  e.  B  <->  E. x  e.  B  x  =  A )
 
Theoremhbral 2403 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 2404  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 2405  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
 
Theoremnfraldxy 2406* Not-free for restricted universal quantification where  x and  y are distinct. See nfraldya 2408 for a version with  y and  A distinct instead. (Contributed by Jim Kingdon, 29-May-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y  e.  A  ps )
 
Theoremnfrexdxy 2407* Not-free for restricted existential quantification where  x and  y are distinct. See nfrexdya 2409 for a version with  y and  A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y  e.  A  ps )
 
Theoremnfraldya 2408* Not-free for restricted universal quantification where  y and  A are distinct. See nfraldxy 2406 for a version with  x and  y distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y  e.  A  ps )
 
Theoremnfrexdya 2409* Not-free for restricted existential quantification where  y and  A are distinct. See nfrexdxy 2407 for a version with  x and  y distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y  e.  A  ps )
 
Theoremnfralxy 2410* Not-free for restricted universal quantification where  x and  y are distinct. See nfralya 2412 for a version with  y and 
A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x A. y  e.  A  ph
 
Theoremnfrexxy 2411* Not-free for restricted existential quantification where  x and  y are distinct. See nfrexya 2413 for a version with  y and 
A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x E. y  e.  A  ph
 
Theoremnfralya 2412* Not-free for restricted universal quantification where  y and  A are distinct. See nfralxy 2410 for a version with  x and  y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x A. y  e.  A  ph
 
Theoremnfrexya 2413* Not-free for restricted existential quantification where  y and  A are distinct. See nfrexxy 2411 for a version with  x and  y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x E. y  e.  A  ph
 
Theoremnfra2xy 2414* Not-free given two restricted quantifiers. (Contributed by Jim Kingdon, 20-Aug-2018.)
 |- 
 F/ y A. x  e.  A  A. y  e.  B  ph
 
Theoremnfre1 2415  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 2416* 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 2417 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
 |-  ( A. x ph  ->  A. x  e.  A  ph )
 
Theoremrexex 2418 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
 |-  ( E. x  e.  A  ph  ->  E. x ph )
 
Theoremrsp 2419 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
 |-  ( A. x  e.  A  ph  ->  ( x  e.  A  ->  ph )
 )
 
Theoremrspe 2420 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
 |-  ( ( x  e.  A  /\  ph )  ->  E. x  e.  A  ph )
 
Theoremrsp2 2421 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 2422 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 2423 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
 |- 
 A. x  e.  A  ph   =>    |-  ( x  e.  A  -> 
 ph )
 
Theoremrgen 2424 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
 |-  ( x  e.  A  -> 
 ph )   =>    |- 
 A. x  e.  A  ph
 
Theoremrgen2a 2425* Generalization rule for restricted quantification. Note that  x and  y needn't be distinct (and illustrates the use of dvelimor 1939). (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.)
 |-  ( ( x  e.  A  /\  y  e.  A )  ->  ph )   =>    |-  A. x  e.  A  A. y  e.  A  ph
 
Theoremrgenw 2426 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
 |-  ph   =>    |- 
 A. x  e.  A  ph
 
Theoremrgen2w 2427 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 2428 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
 |-  ( A. x  e.  A  ph  ->  ps )   &    |-  ( x  e.  A  ->  ph )   =>    |- 
 ps
 
Theoremmprgbir 2429 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 2430 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 2431 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 2432 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 2433 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 2434 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 2435 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 2436 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 2437* 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 2438* 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 2439* 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 2440 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 2441* 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 2442* 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 2443* 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 2444 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 2445 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 2446* 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 2447 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 2448* 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 2449* 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 2450* 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 2451* 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 2452* 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 2453* 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 2454* 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 2455* 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 2456* 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 2457 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 2458 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 2459 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 2460 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 2461 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
 |-  ( x  e.  A  ->  -.  ps )   =>    |-  -.  E. x  e.  A  ps
 
Theoremnrexdv 2462* 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 2463 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 2464 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 2465 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 2466 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 2467 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 2468* 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 2469* 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 2470* 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 2471* 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 ) )
 
Theoremreximddv 2472* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  ( ( ph  /\  ( x  e.  A  /\  ps ) )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  ps )   =>    |-  ( ph  ->  E. x  e.  A  ch )
 
Theoremreximddv2 2473* Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  ( ( ( (
 ph  /\  x  e.  A )  /\  y  e.  B )  /\  ps )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )   =>    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ch )
 
Theoremr19.12 2474* 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 2475 Closed theorem form of r19.23 2476. (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 2476 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 2477* 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 2478 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 2479* 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 2480* 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 2481* Weaker version of rexlimiv 2479. (Contributed by FL, 19-Sep-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x  e.  A  ph 
 ->  ps )
 
Theoremrexlimd 2482 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 2483 Version of rexlimd 2482 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 2484* 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 2485* 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 2486* 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 2487* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2484. (Contributed by NM, 7-Jun-2015.)
 |-  ( ( ph  /\  x  e.  A  /\  ps )  ->  ch )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
 
Theoremrexlimdvw 2488* 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 2489* 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 2490* 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 2491* 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 2492* 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 2493 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 2494 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 2495 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 2496 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 2497 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 ) )
 
Theoremrexbi 2498 Distribute a restricted existential quantifier over a biconditional. Theorem 19.18 of [Margaris] p. 90 with restricted quantification. (Contributed by Jim Kingdon, 21-Jan-2019.)
 |-  ( A. x  e.  A  ( ph  <->  ps )  ->  ( E. x  e.  A  ph  <->  E. x  e.  A  ps ) )
 
Theoremralbiim 2499 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 2500* 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 )
 )
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