P. 26 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2501-2600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2ralbidv 2501*	Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) )
Theorem	2rexbidv 2502*	Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) )
Theorem	rexralbidv 2503*	Formula-building rule for restricted quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. A A. y e. B ps <-> E. x e. A A. y e. B ch ) )
Theorem	ralinexa 2504	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 ) )
Theorem	risset 2505*	Two ways to say " A belongs to B". (Contributed by NM, 22-Nov-1994.)
|- ( A e. B <-> E. x e. B x = A )
Theorem	hbral 2506	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 )
Theorem	hbra1 2507	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 )
Theorem	nfra1 2508	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
Theorem	nfraldw 2509*	Not-free for restricted universal quantification where x and y are distinct. See nfraldya 2512 for a version with y and A distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x A. y e. A ps )
Theorem	nfraldxy 2510*	Old name for nfraldw 2509. (Contributed by Jim Kingdon, 29-May-2018.) (New usage is discouraged.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x A. y e. A ps )
Theorem	nfrexdxy 2511*	Not-free for restricted existential quantification where x and y are distinct. See nfrexdya 2513 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 )
Theorem	nfraldya 2512*	Not-free for restricted universal quantification where y and A are distinct. See nfraldxy 2510 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 )
Theorem	nfrexdya 2513*	Not-free for restricted existential quantification where y and A are distinct. See nfrexdxy 2511 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 )
Theorem	nfralw 2514*	Bound-variable hypothesis builder for restricted quantification. See nfralya 2517 for a version with y and A distinct instead of x and y. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x A. y e. A ph
Theorem	nfralxy 2515*	Old name for nfralw 2514. (Contributed by Jim Kingdon, 30-May-2018.) (New usage is discouraged.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x A. y e. A ph
Theorem	nfrexxy 2516*	Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2518 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
Theorem	nfralya 2517*	Not-free for restricted universal quantification where y and A are distinct. See nfralxy 2515 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
Theorem	nfrexya 2518*	Not-free for restricted existential quantification where y and A are distinct. See nfrexxy 2516 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
Theorem	nfra2xy 2519*	Not-free given two restricted quantifiers. (Contributed by Jim Kingdon, 20-Aug-2018.)
|- F/ y A. x e. A A. y e. B ph
Theorem	nfre1 2520	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
Theorem	r3al 2521*	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 ) )
Theorem	alral 2522	Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
|- ( A. x ph -> A. x e. A ph )
Theorem	rexex 2523	Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
|- ( E. x e. A ph -> E. x ph )
Theorem	rsp 2524	Restricted specialization. (Contributed by NM, 17-Oct-1996.)
|- ( A. x e. A ph -> ( x e. A -> ph ) )
Theorem	rspa 2525	Restricted specialization. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A. x e. A ph /\ x e. A ) -> ph )
Theorem	rspe 2526	Restricted specialization. (Contributed by NM, 12-Oct-1999.)
|- ( ( x e. A /\ ph ) -> E. x e. A ph )
Theorem	rsp2 2527	Restricted specialization. (Contributed by NM, 11-Feb-1997.)
|- ( A. x e. A A. y e. B ph -> ( ( x e. A /\ y e. B ) -> ph ) )
Theorem	rsp2e 2528	Restricted specialization. (Contributed by FL, 4-Jun-2012.)
|- ( ( x e. A /\ y e. B /\ ph ) -> E. x e. A E. y e. B ph )
Theorem	rspec 2529	Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
|- A. x e. A ph   =>    |- ( x e. A -> ph )
Theorem	rgen 2530	Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
|- ( x e. A -> ph )   =>    |- A. x e. A ph
Theorem	rgen2a 2531*	Generalization rule for restricted quantification. Note that x and y are not required to be disjoint. This proof illustrates the use of dvelim 2017. Usage of rgen2 2563 instead is highly encouraged. (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.) (New usage is discouraged.)
|- ( ( x e. A /\ y e. A ) -> ph )   =>    |- A. x e. A A. y e. A ph
Theorem	rgenw 2532	Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
|- ph   =>    |- A. x e. A ph
Theorem	rgen2w 2533	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
Theorem	mprg 2534	Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
|- ( A. x e. A ph -> ps )   &    |- ( x e. A -> ph )   =>    |- ps
Theorem	mprgbir 2535	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
Theorem	ralim 2536	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 ) )
Theorem	ralimi2 2537	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 )
Theorem	ralimia 2538	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 )
Theorem	ralimiaa 2539	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 )
Theorem	ralimi 2540	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 )
Theorem	2ralimi 2541	Inference quantifying both antecedent and consequent two times, with strong hypothesis. (Contributed by AV, 3-Dec-2021.)
|- ( ph -> ps )   =>    |- ( A. x e. A A. y e. B ph -> A. x e. A A. y e. B ps )
Theorem	ral2imi 2542	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 ) )
Theorem	ralimdaa 2543	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 ) )
Theorem	ralimdva 2544*	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 ) )
Theorem	ralimdv 2545*	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 ) )
Theorem	ralimdvva 2546*	Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1457). (Contributed by AV, 27-Nov-2019.)
|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x e. A A. y e. B ps -> A. x e. A A. y e. B ch ) )
Theorem	ralimdv2 2547*	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 ) )
Theorem	ralrimi 2548	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 )
Theorem	ralrimiv 2549*	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 )
Theorem	ralrimiva 2550*	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 )
Theorem	ralrimivw 2551*	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 )
Theorem	r19.21t 2552	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 ) ) )
Theorem	r19.21 2553	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 ) )
Theorem	r19.21v 2554*	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 ) )
Theorem	ralrimd 2555	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 ) )
Theorem	ralrimdv 2556*	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 ) )
Theorem	ralrimdva 2557*	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 ) )
Theorem	ralrimivv 2558*	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 )
Theorem	ralrimivva 2559*	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 )
Theorem	ralrimivvva 2560*	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 )
Theorem	ralrimdvv 2561*	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 ) )
Theorem	ralrimdvva 2562*	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 ) )
Theorem	rgen2 2563*	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
Theorem	rgen3 2564*	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
Theorem	r19.21bi 2565	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 )
Theorem	rspec2 2566	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 )
Theorem	rspec3 2567	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 )
Theorem	r19.21be 2568	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 )
Theorem	nrex 2569	Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
|- ( x e. A -> -. ps )   =>    |- -. E. x e. A ps
Theorem	nrexdv 2570*	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 )
Theorem	rexim 2571	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 ) )
Theorem	reximia 2572	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 )
Theorem	reximi2 2573	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 )
Theorem	reximi 2574	Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
|- ( ph -> ps )   =>    |- ( E. x e. A ph -> E. x e. A ps )
Theorem	reximdai 2575	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 ) )
Theorem	reximdv2 2576*	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 ) )
Theorem	reximdvai 2577*	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 ) )
Theorem	reximdv 2578*	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 ) )
Theorem	reximdva 2579*	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 ) )
Theorem	reximddv 2580*	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 )
Theorem	reximssdv 2581*	Derivation of a restricted existential quantification over a subset (the second hypothesis implies A C_ B), deduction form. (Contributed by AV, 21-Aug-2022.)
|- ( ph -> E. x e. B ps )   &    |- ( ( ph /\ ( x e. B /\ ps ) ) -> x e. A )   &    |- ( ( ph /\ ( x e. B /\ ps ) ) -> ch )   =>    |- ( ph -> E. x e. A ch )
Theorem	reximddv2 2582*	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 )
Theorem	r19.12 2583*	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 )
Theorem	r19.23t 2584	Closed theorem form of r19.23 2585. (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 ) ) )
Theorem	r19.23 2585	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 ) )
Theorem	r19.23v 2586*	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 ) )
Theorem	rexlimi 2587	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 )
Theorem	rexlimiv 2588*	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 )
Theorem	rexlimiva 2589*	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 )
Theorem	rexlimivw 2590*	Weaker version of rexlimiv 2588. (Contributed by FL, 19-Sep-2011.)
|- ( ph -> ps )   =>    |- ( E. x e. A ph -> ps )
Theorem	rexlimd 2591	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 ) )
Theorem	rexlimd2 2592	Version of rexlimd 2591 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 ) )
Theorem	rexlimdv 2593*	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 ) )
Theorem	rexlimdva 2594*	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 ) )
Theorem	rexlimdvaa 2595*	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 ) )
Theorem	rexlimdv3a 2596*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2593. (Contributed by NM, 7-Jun-2015.)
|- ( ( ph /\ x e. A /\ ps ) -> ch )   =>    |- ( ph -> ( E. x e. A ps -> ch ) )
Theorem	rexlimdva2 2597*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )   =>    |- ( ph -> ( E. x e. A ps -> ch ) )
Theorem	rexlimdvw 2598*	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 ) )
Theorem	rexlimddv 2599*	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 )
Theorem	rexlimivv 2600*	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 )