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Theorem List for Intuitionistic Logic Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremralrimivva 2401* 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 2402* 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 2403* 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 2404* 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 2405* 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 2406* 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 2407 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 2408 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 2409 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 2410 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 2411 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
 |-  ( x  e.  A  ->  -.  ps )   =>    |-  -.  E. x  e.  A  ps
 
Theoremnrexdv 2412* 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 2413 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 2414 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 2415 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 2416 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 2417 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 2418* 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 2419* 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 2420* 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 2421* 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 2422* 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 2423 Closed theorem form of r19.23 2424. (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 2424 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 2425* 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 2426 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 2427* 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 2428* 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 2429* Weaker version of rexlimiv 2427. (Contributed by FL, 19-Sep-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x  e.  A  ph 
 ->  ps )
 
Theoremrexlimd 2430 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 2431 Version of rexlimd 2430 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 2432* 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 2433* 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 2434* 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 2435* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2432. (Contributed by NM, 7-Jun-2015.)
 |-  ( ( ph  /\  x  e.  A  /\  ps )  ->  ch )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
 
Theoremrexlimdvw 2436* 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 2437* 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 2438* 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 2439* 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 2440* 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 2441 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 2442 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 2443 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 2444 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 2445 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 2446 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 2447 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 2448* 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 2449* 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 2450 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 2451 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 2452 A commonly used pattern based on r19.29 2450 (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 2453* A commonly used pattern based on r19.29 2450 (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 2454* A commonly used pattern based on r19.29 2450 (Contributed by Thierry Arnoux, 22-Nov-2017.)
 |-  ( ( ( ph  /\  x  e.  A ) 
 /\  ps )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  ps )   =>    |-  ( ph  ->  ch )
 
Theoremr19.29d2r 2455 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 ) )
 
Theoremr19.29vva 2456* A commonly used pattern based on r19.29 2450, 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 )
 
Theoremr19.32r 2457 One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence. (Contributed by Jim Kingdon, 19-Aug-2018.)
 |- 
 F/ x ph   =>    |-  ( ( ph  \/  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  \/  ps )
 )
 
Theoremr19.32vr 2458* One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence, as seen at r19.32vdc 2459. (Contributed by Jim Kingdon, 19-Aug-2018.)
 |-  ( ( ph  \/  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  \/  ps )
 )
 
Theoremr19.32vdc 2459* Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, where  ph is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
 |-  (DECID 
 ph  ->  ( A. x  e.  A  ( ph  \/  ps )  <->  ( ph  \/  A. x  e.  A  ps ) ) )
 
Theoremr19.35-1 2460 Restricted quantifier version of 19.35-1 1515. (Contributed by Jim Kingdon, 4-Jun-2018.)
 |-  ( E. x  e.  A  ( ph  ->  ps )  ->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )
 
Theoremr19.36av 2461* One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. In classical logic, the converse would hold if  A has at least one element, but in intuitionistic logic, that is not a sufficient condition. (Contributed by NM, 22-Oct-2003.)
 |-  ( E. x  e.  A  ( ph  ->  ps )  ->  ( A. x  e.  A  ph  ->  ps ) )
 
Theoremr19.37 2462 Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. In classical logic the converse would hold if  A has at least one element, but that is not sufficient in intuitionistic logic. (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 2463* Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
 |-  ( E. x  e.  A  ( ph  ->  ps )  ->  ( ph  ->  E. x  e.  A  ps ) )
 
Theoremr19.40 2464 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 2465 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 2466* 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 2467* 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 2468 Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)
 |-  ( E. x  e.  A  ( ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
 
Theoremr19.44av 2469* 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 2470* 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 2471* 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 2472* 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 2473* 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 2474* 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 2475* 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 2476* 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 )
 
Theoremralcom3 2477 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 2478* 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 2479* 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 2480* 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 ) )
 
Theoremnfreu1 2481  x is not free in  E! x  e.  A ph. (Contributed by NM, 19-Mar-1997.)
 |- 
 F/ x E! x  e.  A  ph
 
Theoremnfrmo1 2482  x is not free in  E* x  e.  A ph. (Contributed by NM, 16-Jun-2017.)
 |- 
 F/ x E* x  e.  A  ph
 
Theoremnfreudxy 2483* Not-free deduction for restricted uniqueness. This is a version where  x and  y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E! y  e.  A  ps )
 
Theoremnfreuxy 2484* Not-free for restricted uniqueness. This is a version where  x and  y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x E! y  e.  A  ph
 
Theoremrabid 2485 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 2486* 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 2487 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2548. (Contributed by NM, 25-Nov-2013.)
 |-  ( A. x  e.  A  ( ps  <->  ch )  <->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch } )
 
Theoremrabswap 2488 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 2489 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
 |-  F/_ x { x  e.  A  |  ph }
 
Theoremnfrabxy 2490* A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)
 |- 
 F/ x ph   &    |-  F/_ x A   =>    |-  F/_ x { y  e.  A  |  ph }
 
Theoremreubida 2491 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 2492* 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 2493* 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 2494 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 2495 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 2496 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 2497* 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 2498* 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 2499 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 2500 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 )
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