P. 28 of Theorem List - New Foundations Explorer

Theorem List for New Foundations Explorer - 2701-2800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	r19.21 2701	Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)
|- F/xph   =>   |- (A.x e. A (ph -> ps) <-> (ph -> A.x e. A ps))
Theorem	r19.21v 2702*	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 2703	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)
|- F/xph   &   |- F/xps   &   |- (ph -> (ps -> (x e. A -> ch)))   =>   |- (ph -> (ps -> A.x e. A ch))
Theorem	ralrimdv 2704*	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 2705*	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 2706*	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 2707*	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 2708*	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 2709*	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 2710*	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 2711*	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 2712*	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 2713	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 2714	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 2715	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 2716	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 2717	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 2718*	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 2719	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 2720	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 2721	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 2722	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 2723	Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)
|- F/xph   &   |- (ph -> (x e. A -> (ps -> ch)))   =>   |- (ph -> (E.x e. A ps -> E.x e. A ch))
Theorem	reximdv2 2724*	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 2725*	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 2726*	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 2727*	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	r19.12 2728*	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 2729	Closed theorem form of r19.23 2730. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
|- ( F/xps -> (A.x e. A (ph -> ps) <-> (E.x e. A ph -> ps)))
Theorem	r19.23 2730	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/xps   =>   |- (A.x e. A (ph -> ps) <-> (E.x e. A ph -> ps))
Theorem	r19.23v 2731*	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 2732	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/xps   &   |- (x e. A -> (ph -> ps))   =>   |- (E.x e. A ph -> ps)
Theorem	rexlimiv 2733*	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 2734*	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 2735*	Weaker version of rexlimiv 2733. (Contributed by FL, 19-Sep-2011.)
|- (ph -> ps)   =>   |- (E.x e. A ph -> ps)
Theorem	rexlimd 2736	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/xph   &   |- F/xch   &   |- (ph -> (x e. A -> (ps -> ch)))   =>   |- (ph -> (E.x e. A ps -> ch))
Theorem	rexlimd2 2737	Version of rexlimd 2736 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
|- F/xph   &   |- (ph -> F/xch)   &   |- (ph -> (x e. A -> (ps -> ch)))   =>   |- (ph -> (E.x e. A ps -> ch))
Theorem	rexlimdv 2738*	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 2739*	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 2740*	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 2741*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2738. (Contributed by NM, 7-Jun-2015.)
|- ((ph /\ x e. A /\ ps) -> ch)   =>   |- (ph -> (E.x e. A ps -> ch))
Theorem	rexlimdvw 2742*	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 2743*	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 2744*	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 2745*	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 2746*	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 2747	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	r19.26-2 2748	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 2749	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 2750	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 2751	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	ralbiim 2752	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 2753*	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 2754*	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 2755	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 2756	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	r19.30 2757	Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 25-Feb-2011.)
|- (A.x e. A (ph \/ ps) -> (A.x e. A ph \/ E.x e. A ps))
Theorem	r19.32v 2758*	Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 25-Nov-2003.)
|- (A.x e. A (ph \/ ps) <-> (ph \/ A.x e. A ps))
Theorem	r19.35 2759	Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 20-Sep-2003.)
|- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))
Theorem	r19.36av 2760*	One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. The other direction doesn't hold when A is empty. (Contributed by NM, 22-Oct-2003.)
|- (E.x e. A (ph -> ps) -> (A.x e. A ph -> ps))
Theorem	r19.37 2761	Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
|- F/xph   =>   |- (E.x e. A (ph -> ps) -> (ph -> E.x e. A ps))
Theorem	r19.37av 2762*	Restricted version of one direction of Theorem 19.37 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	r19.40 2763	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 2764	Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
|- F/xps   =>   |- (E.x e. A (ph /\ ps) <-> (E.x e. A ph /\ ps))
Theorem	r19.41v 2765*	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 2766*	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 2767	Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
|- (E.x e. A (ph \/ ps) <-> (E.x e. A ph \/ E.x e. A ps))
Theorem	r19.44av 2768*	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 2769*	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 2770*	Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- F/_yA   &   |- F/_xB   =>   |- (A.x e. A A.y e. B ph <-> A.y e. B A.x e. A ph)
Theorem	rexcomf 2771*	Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- F/_yA   &   |- F/_xB   =>   |- (E.x e. A E.y e. B ph <-> E.y e. B E.x e. A ph)
Theorem	ralcom 2772*	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 2773*	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	rexcom13 2774*	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 2775*	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)
Theorem	ralcom2 2776*	Commutation of restricted quantifiers. Note that x and y needn't be distinct (this makes the proof longer). (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)
|- (A.x e. A A.y e. A ph -> A.y e. A A.x e. A ph)
Theorem	ralcom3 2777	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))
Theorem	reean 2778*	Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
|- F/yph   &   |- F/xps   =>   |- (E.x e. A E.y e. B (ph /\ ps) <-> (E.x e. A ph /\ E.y e. B ps))
Theorem	reeanv 2779*	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))
Theorem	3reeanv 2780*	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))
Theorem	2ralor 2781*	Distribute quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.)
|- (A.x e. A A.y e. B (ph \/ ps) <-> (A.x e. A ph \/ A.y e. B ps))
Theorem	nfreu1 2782	x is not free in E!x e. Aph. (Contributed by NM, 19-Mar-1997.)
|- F/xE!x e. A ph
Theorem	nfrmo1 2783	x is not free in E*x e. Aph. (Contributed by NM, 16-Jun-2017.)
|- F/xE*x e. A ph
Theorem	nfreud 2784	Deduction version of nfreu 2786. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
|- F/yph   &   |- (ph -> F/_xA)   &   |- (ph -> F/xps)   =>   |- (ph -> F/xE!y e. A ps)
Theorem	nfrmod 2785	Deduction version of nfrmo 2787. (Contributed by NM, 17-Jun-2017.)
|- F/yph   &   |- (ph -> F/_xA)   &   |- (ph -> F/xps)   =>   |- (ph -> F/xE*y e. A ps)
Theorem	nfreu 2786	Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.)
|- F/_xA   &   |- F/xph   =>   |- F/xE!y e. A ph
Theorem	nfrmo 2787	Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 16-Jun-2017.)
|- F/_xA   &   |- F/xph   =>   |- F/xE*y e. A ph
Theorem	rabid 2788	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))
Theorem	rabid2 2789*	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)
Theorem	rabbi 2790	Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2851. (Contributed by NM, 25-Nov-2013.)
|- (A.x e. A (ps <-> ch) <-> {x e. A | ps} = {x e. A | ch})
Theorem	rabswap 2791	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}
Theorem	nfrab1 2792	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
|- F/_x{x e. A | ph}
Theorem	nfrab 2793	A variable not free in a wff remains so in a restricted class abstraction. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.)
|- F/xph   &   |- F/_xA   =>   |- F/_x{y e. A | ph}
Theorem	reubida 2794	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)
|- F/xph   &   |- ((ph /\ x e. A) -> (ps <-> ch))   =>   |- (ph -> (E!x e. A ps <-> E!x e. A ch))
Theorem	reubidva 2795*	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))
Theorem	reubidv 2796*	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))
Theorem	reubiia 2797	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)
Theorem	reubii 2798	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)
Theorem	rmobida 2799	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
|- F/xph   &   |- ((ph /\ x e. A) -> (ps <-> ch))   =>   |- (ph -> (E*x e. A ps <-> E*x e. A ch))
Theorem	rmobidva 2800*	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))