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Theorem List for Intuitionistic Logic Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremralimi 2401 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 2402 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 2403 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 2404* 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 2405* 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 2406* 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 2407 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 2408* 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 2409* 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 2410* 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 2411 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 2412 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 2413* 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 2414 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 2415* 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 2416* 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 2417* 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 2418* 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 2419* 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 2420* 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 2421* 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 2422* 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 2423* 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 2424 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 2425 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 2426 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 2427 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 2428 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
 |-  ( x  e.  A  ->  -.  ps )   =>    |-  -.  E. x  e.  A  ps
 
Theoremnrexdv 2429* 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 2430 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 2431 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 2432 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 2433 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 2434 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 2435* 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 2436* 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 2437* 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 2438* 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 2439* 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 2440 Closed theorem form of r19.23 2441. (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 2441 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 2442* 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 2443 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 2444* 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 2445* 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 2446* Weaker version of rexlimiv 2444. (Contributed by FL, 19-Sep-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x  e.  A  ph 
 ->  ps )
 
Theoremrexlimd 2447 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 2448 Version of rexlimd 2447 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 2449* 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 2450* 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 2451* 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 2452* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2449. (Contributed by NM, 7-Jun-2015.)
 |-  ( ( ph  /\  x  e.  A  /\  ps )  ->  ch )   =>    |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
 
Theoremrexlimdvw 2453* 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 2454* 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 2455* 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 2456* 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 2457* 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 2458 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 2459 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 2460 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 2461 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 2462 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 2463 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 2464 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 2465* 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 2466* 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 2467 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 2468 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 2469 A commonly used pattern based on r19.29 2467 (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 2470* A commonly used pattern based on r19.29 2467 (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 2471* A commonly used pattern based on r19.29 2467 (Contributed by Thierry Arnoux, 22-Nov-2017.)
 |-  ( ( ( ph  /\  x  e.  A ) 
 /\  ps )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  ps )   =>    |-  ( ph  ->  ch )
 
Theoremr19.29d2r 2472 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 2473* A commonly used pattern based on r19.29 2467, 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 2474 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 2475* 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 2476. (Contributed by Jim Kingdon, 19-Aug-2018.)
 |-  ( ( ph  \/  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  \/  ps )
 )
 
Theoremr19.32vdc 2476* 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 2477 Restricted quantifier version of 19.35-1 1531. (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 2478* 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 2479 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 2480* 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 2481 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 2482 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 2483* 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 2484* 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 2485 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 2486* 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 2487* 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 2488* 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 2489* 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 2490* 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 2491* 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 2492* 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 2493* 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 2494 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 2495* 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 2496* 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 2497* 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 2498  x is not free in  E! x  e.  A ph. (Contributed by NM, 19-Mar-1997.)
 |- 
 F/ x E! x  e.  A  ph
 
Theoremnfrmo1 2499  x is not free in  E* x  e.  A ph. (Contributed by NM, 16-Jun-2017.)
 |- 
 F/ x E* x  e.  A  ph
 
Theoremnfreudxy 2500* 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 )
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