P. 25 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2401-2500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ralrimivva 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 )
Theorem	ralrimivvva 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 )
Theorem	ralrimdvv 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 ) )
Theorem	ralrimdvva 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 ) )
Theorem	rgen2 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
Theorem	rgen3 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
Theorem	r19.21bi 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 )
Theorem	rspec2 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 )
Theorem	rspec3 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 )
Theorem	r19.21be 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 )
Theorem	nrex 2411	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 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 )
Theorem	rexim 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 ) )
Theorem	reximia 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 )
Theorem	reximi2 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 )
Theorem	reximi 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 )
Theorem	reximdai 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 ) )
Theorem	reximdv2 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 ) )
Theorem	reximdvai 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 ) )
Theorem	reximdv 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 ) )
Theorem	reximdva 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 ) )
Theorem	r19.12 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 )
Theorem	r19.23t 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 ) ) )
Theorem	r19.23 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 ) )
Theorem	r19.23v 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 ) )
Theorem	rexlimi 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 )
Theorem	rexlimiv 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 )
Theorem	rexlimiva 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 )
Theorem	rexlimivw 2429*	Weaker version of rexlimiv 2427. (Contributed by FL, 19-Sep-2011.)
|- ( ph -> ps )   =>    |- ( E. x e. A ph -> ps )
Theorem	rexlimd 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 ) )
Theorem	rexlimd2 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 ) )
Theorem	rexlimdv 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 ) )
Theorem	rexlimdva 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 ) )
Theorem	rexlimdvaa 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 ) )
Theorem	rexlimdv3a 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 ) )
Theorem	rexlimdvw 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 ) )
Theorem	rexlimddv 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 )
Theorem	rexlimivv 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 )
Theorem	rexlimdvv 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 ) )
Theorem	rexlimdvva 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 ) )
Theorem	r19.26 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 ) )
Theorem	r19.26-2 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 ) )
Theorem	r19.26-3 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 ) )
Theorem	r19.26m 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 ) )
Theorem	ralbi 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 ) )
Theorem	rexbi 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 ) )
Theorem	ralbiim 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 ) ) )
Theorem	r19.27av 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 ) )
Theorem	r19.28av 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 ) )
Theorem	r19.29 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 ) )
Theorem	r19.29r 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 ) )
Theorem	r19.29af2 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 )
Theorem	r19.29af 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 )
Theorem	r19.29a 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 )
Theorem	r19.29d2r 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 ) )
Theorem	r19.29vva 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 )
Theorem	r19.32r 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 ) )
Theorem	r19.32vr 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 ) )
Theorem	r19.32vdc 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 ) ) )
Theorem	r19.35-1 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 ) )
Theorem	r19.36av 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 ) )
Theorem	r19.37 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 ) )
Theorem	r19.37av 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 ) )
Theorem	r19.40 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 ) )
Theorem	r19.41 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 ) )
Theorem	r19.41v 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 ) )
Theorem	r19.42v 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 ) )
Theorem	r19.43 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 ) )
Theorem	r19.44av 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 ) )
Theorem	r19.45av 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 ) )
Theorem	ralcomf 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 )
Theorem	rexcomf 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 )
Theorem	ralcom 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 )
Theorem	rexcom 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 )
Theorem	rexcom13 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 )
Theorem	rexrot4 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 )
Theorem	ralcom3 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 ) )
Theorem	reean 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 ) )
Theorem	reeanv 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 ) )
Theorem	3reeanv 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 ) )
Theorem	nfreu1 2481	x is not free in E! x e. A ph. (Contributed by NM, 19-Mar-1997.)
|- F/ x E! x e. A ph
Theorem	nfrmo1 2482	x is not free in E* x e. A ph. (Contributed by NM, 16-Jun-2017.)
|- F/ x E* x e. A ph
Theorem	nfreudxy 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 )
Theorem	nfreuxy 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
Theorem	rabid 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 ) )
Theorem	rabid2 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 )
Theorem	rabbi 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 } )
Theorem	rabswap 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 }
Theorem	nfrab1 2489	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
|- F/_ x { x e. A | ph }
Theorem	nfrabxy 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 }
Theorem	reubida 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 ) )
Theorem	reubidva 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 ) )
Theorem	reubidv 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 ) )
Theorem	reubiia 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 )
Theorem	reubii 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 )
Theorem	rmobida 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 ) )
Theorem	rmobidva 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 ) )
Theorem	rmobidv 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 ) )
Theorem	rmobiia 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 )
Theorem	rmobii 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 )