P. 27 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2601-2700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ralimdvva 2601*	Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1505). (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 2602*	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 2603	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 2604*	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 2605*	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 2606*	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 2607	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 2608	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 2609*	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 2610	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 2611*	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 2612*	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 2613*	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 2614*	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 2615*	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 2616*	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 2617*	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 2618*	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 2619*	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 2620	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 2621	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 2622	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 2623	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 2624	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 2625*	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 2626	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 2627	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 2628	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 2629	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 2630	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 2631*	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 2632*	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 2633*	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 2634*	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 2635*	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 2636*	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 2637*	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	rexanaliim 2638	A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Revised by Jim Kingdon, 18-Jan-2026.)
|- ( E. x e. A ( ph /\ -. ps ) -> -. A. x e. A ( ph -> ps ) )
Theorem	r19.12 2639*	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 2640	Closed theorem form of r19.23 2641. (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 2641	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 2642*	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 2643	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 2644*	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 2645*	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 2646*	Weaker version of rexlimiv 2644. (Contributed by FL, 19-Sep-2011.)
|- ( ph -> ps )   =>    |- ( E. x e. A ph -> ps )
Theorem	rexlimd 2647	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 2648	Version of rexlimd 2647 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 2649*	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 2650*	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 2651*	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 2652*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2649. (Contributed by NM, 7-Jun-2015.)
|- ( ( ph /\ x e. A /\ ps ) -> ch )   =>    |- ( ph -> ( E. x e. A ps -> ch ) )
Theorem	rexlimdva2 2653*	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 2654*	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 2655*	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 2656*	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 )
Theorem	rexlimdvv 2657*	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 ) )
Theorem	rexlimdvva 2658*	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 ) )
Theorem	r19.26 2659	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 ) )
Theorem	r19.27v 2660*	Restricted quantitifer version of one direction of 19.27 1609. (The other direction holds when A is inhabited, see r19.27mv 3591.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
|- ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) )
Theorem	r19.28v 2661*	Restricted quantifier version of one direction of 19.28 1611. (The other direction holds when A is inhabited, see r19.28mv 3587.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
|- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) )
Theorem	r19.26-2 2662	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 ) )
Theorem	r19.26-3 2663	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 ) )
Theorem	r19.26m 2664	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 ) )
Theorem	ralbi 2665	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 ) )
Theorem	rexbi 2666	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 ) )
Theorem	ralbiim 2667	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 ) ) )
Theorem	r19.27av 2668*	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 ) )
Theorem	r19.28av 2669*	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 ) )
Theorem	r19.29 2670	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 ) )
Theorem	r19.29r 2671	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 ) )
Theorem	ralnex2 2672	Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.)
|- ( A. x e. A A. y e. B -. ph <-> -. E. x e. A E. y e. B ph )
Theorem	r19.29af2 2673	A commonly used pattern based on r19.29 2670. (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 )
Theorem	r19.29af 2674*	A commonly used pattern based on r19.29 2670. (Contributed by Thierry Arnoux, 29-Nov-2017.)
|- F/ x ph   &    |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )   &    |- ( ph -> E. x e. A ps )   =>    |- ( ph -> ch )
Theorem	r19.29an 2675*	A commonly used pattern based on r19.29 2670. (Contributed by Thierry Arnoux, 29-Dec-2019.)
|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )   =>    |- ( ( ph /\ E. x e. A ps ) -> ch )
Theorem	r19.29a 2676*	A commonly used pattern based on r19.29 2670. (Contributed by Thierry Arnoux, 22-Nov-2017.)
|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )   &    |- ( ph -> E. x e. A ps )   =>    |- ( ph -> ch )
Theorem	r19.29d2r 2677	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 ) )
Theorem	r19.29vva 2678*	A commonly used pattern based on r19.29 2670, 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 )
Theorem	r19.32r 2679	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 ) )
Theorem	r19.30dc 2680	Restricted quantifier version of 19.30dc 1675. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
|- ( ( A. x e. A ( ph \/ ps ) /\ DECID E. x e. A ps ) -> ( A. x e. A ph \/ E. x e. A ps ) )
Theorem	r19.32vr 2681*	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 2682. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- ( ( ph \/ A. x e. A ps ) -> A. x e. A ( ph \/ ps ) )
Theorem	r19.32vdc 2682*	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 ) ) )
Theorem	r19.35-1 2683	Restricted quantifier version of 19.35-1 1672. (Contributed by Jim Kingdon, 4-Jun-2018.)
|- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> E. x e. A ps ) )
Theorem	r19.36av 2684*	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 ) )
Theorem	r19.37 2685	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 ) )
Theorem	r19.37av 2686*	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 ) )
Theorem	r19.40 2687	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 ) )
Theorem	r19.41 2688	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 ) )
Theorem	r19.41v 2689*	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 ) )
Theorem	r19.42v 2690*	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 ) )
Theorem	r19.43 2691	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 ) )
Theorem	r19.44av 2692*	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 ) )
Theorem	r19.45av 2693*	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 ) )
Theorem	ralcomf 2694*	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 )
Theorem	rexcomf 2695*	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 )
Theorem	ralcom 2696*	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 )
Theorem	rexcom 2697*	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 )
Theorem	ralrot3 2698*	Rotate three restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)
|- ( A. x e. A A. y e. B A. z e. C ph <-> A. z e. C A. x e. A A. y e. B ph )
Theorem	rexcom13 2699*	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 )
Theorem	rexrot4 2700*	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 )