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	nemtbir 2501	An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
|- A =/= B   &    |- ( ph <-> A = B )   =>    |- -. ph
Theorem	nelne1 2502	Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
|- ( ( A e. B /\ -. A e. C ) -> B =/= C )
Theorem	nelne2 2503	Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
|- ( ( A e. C /\ -. B e. C ) -> A =/= B )
Theorem	nelelne 2504	Two classes are different if they don't belong to the same class. (Contributed by Rodolfo Medina, 17-Oct-2010.) (Proof shortened by AV, 10-May-2020.)
|- ( -. A e. B -> ( C e. B -> C =/= A ) )
Theorem	nfne 2505	Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/ x A =/= B
Theorem	nfned 2506	Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/ x A =/= B )
2.1.4.2  Negated membership
Syntax	wnel 2507	Extend wff notation to include negated membership.
wff A e/ B
Definition	df-nel 2508	Define negated membership. (Contributed by NM, 7-Aug-1994.)
|- ( A e/ B <-> -. A e. B )
Theorem	neli 2509	Inference associated with df-nel 2508. (Contributed by BJ, 7-Jul-2018.)
|- A e/ B   =>    |- -. A e. B
Theorem	nelir 2510	Inference associated with df-nel 2508. (Contributed by BJ, 7-Jul-2018.)
|- -. A e. B   =>    |- A e/ B
Theorem	neleq1 2511	Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
|- ( A = B -> ( A e/ C <-> B e/ C ) )
Theorem	neleq2 2512	Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
|- ( A = B -> ( C e/ A <-> C e/ B ) )
Theorem	neleq12d 2513	Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A e/ C <-> B e/ D ) )
Theorem	nfnel 2514	Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/ x A e/ B
Theorem	nfneld 2515	Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/ x A e/ B )
Theorem	elnelne1 2516	Two classes are different if they don't contain the same element. (Contributed by AV, 28-Jan-2020.)
|- ( ( A e. B /\ A e/ C ) -> B =/= C )
Theorem	elnelne2 2517	Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
|- ( ( A e. C /\ B e/ C ) -> A =/= B )
Theorem	nelcon3d 2518	Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)
|- ( ph -> ( A e. B -> C e. D ) )   =>    |- ( ph -> ( C e/ D -> A e/ B ) )
Theorem	elnelall 2519	A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
|- ( A e. B -> ( A e/ B -> ph ) )
2.1.5  Restricted quantification
Syntax	wral 2520	Extend wff notation to include restricted universal quantification.
wff A. x e. A ph
Syntax	wrex 2521	Extend wff notation to include restricted existential quantification.
wff E. x e. A ph
Syntax	wreu 2522	Extend wff notation to include restricted existential uniqueness.
wff E! x e. A ph
Syntax	wrmo 2523	Extend wff notation to include restricted "at most one".
wff E* x e. A ph
Syntax	crab 2524	Extend class notation to include the restricted class abstraction (class builder).
class { x e. A | ph }
Definition	df-ral 2525	Define restricted universal quantification. Special case of Definition 4.15(3) of [TakeutiZaring] p. 22. (Contributed by NM, 19-Aug-1993.)
|- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
Definition	df-rex 2526	Define restricted existential quantification. Special case of Definition 4.15(4) of [TakeutiZaring] p. 22. (Contributed by NM, 30-Aug-1993.)
|- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
Definition	df-reu 2527	Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
|- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
Definition	df-rmo 2528	Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
|- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
Definition	df-rab 2529	Define a restricted class abstraction (class builder), which is the class of all x in A such that ph is true. Definition of [TakeutiZaring] p. 20. (Contributed by NM, 22-Nov-1994.)
|- { x e. A | ph } = { x | ( x e. A /\ ph ) }
Theorem	ralnex 2530	Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
|- ( A. x e. A -. ph <-> -. E. x e. A ph )
Theorem	rexnalim 2531	Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)
|- ( E. x e. A -. ph -> -. A. x e. A ph )
Theorem	nnral 2532	The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1698. (Contributed by Jim Kingdon, 1-Aug-2024.)
|- ( -. -. A. x e. A ph -> A. x e. A -. -. ph )
Theorem	dfrex2dc 2533	Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 29-Jun-2022.)
|- (DECID E. x e. A ph -> ( E. x e. A ph <-> -. A. x e. A -. ph ) )
Theorem	ralexim 2534	Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
|- ( A. x e. A ph -> -. E. x e. A -. ph )
Theorem	rexalim 2535	Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
|- ( E. x e. A ph -> -. A. x e. A -. ph )
Theorem	ralbida 2536	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )
Theorem	rexbida 2537	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )
Theorem	ralbidva 2538*	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 4-Mar-1997.)
|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )
Theorem	rexbidva 2539*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 9-Mar-1997.)
|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )
Theorem	ralbid 2540	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )
Theorem	rexbid 2541	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )
Theorem	ralbidv 2542*	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 20-Nov-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )
Theorem	rexbidv 2543*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 20-Nov-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )
Theorem	ralbidv2 2544*	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997.)
|- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )
Theorem	rexbidv2 2545*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 22-May-1999.)
|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )   =>    |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )
Theorem	ralbid2 2546	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
|- F/ x ph   &    |- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )
Theorem	rexbid2 2547	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
|- F/ x ph   &    |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )   =>    |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )
Theorem	ralbii 2548	Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.)
|- ( ph <-> ps )   =>    |- ( A. x e. A ph <-> A. x e. A ps )
Theorem	rexbii 2549	Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.)
|- ( ph <-> ps )   =>    |- ( E. x e. A ph <-> E. x e. A ps )
Theorem	2ralbii 2550	Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
|- ( ph <-> ps )   =>    |- ( A. x e. A A. y e. B ph <-> A. x e. A A. y e. B ps )
Theorem	2rexbii 2551	Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)
|- ( ph <-> ps )   =>    |- ( E. x e. A E. y e. B ph <-> E. x e. A E. y e. B ps )
Theorem	ralbii2 2552	Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
|- ( ( x e. A -> ph ) <-> ( x e. B -> ps ) )   =>    |- ( A. x e. A ph <-> A. x e. B ps )
Theorem	rexbii2 2553	Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)
|- ( ( x e. A /\ ph ) <-> ( x e. B /\ ps ) )   =>    |- ( E. x e. A ph <-> E. x e. B ps )
Theorem	raleqbii 2554	Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
|- A = B   &    |- ( ps <-> ch )   =>    |- ( A. x e. A ps <-> A. x e. B ch )
Theorem	rexeqbii 2555	Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
|- A = B   &    |- ( ps <-> ch )   =>    |- ( E. x e. A ps <-> E. x e. B ch )
Theorem	ralbiia 2556	Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
|- ( x e. A -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. x e. A ps )
Theorem	rexbiia 2557	Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
|- ( x e. A -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. x e. A ps )
Theorem	2rexbiia 2558*	Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
|- ( ( x e. A /\ y e. B ) -> ( ph <-> ps ) )   =>    |- ( E. x e. A E. y e. B ph <-> E. x e. A E. y e. B ps )
Theorem	r2alf 2559*	Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- F/_ y A   =>    |- ( A. x e. A A. y e. B ph <-> A. x A. y ( ( x e. A /\ y e. B ) -> ph ) )
Theorem	r2exf 2560*	Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- F/_ y A   =>    |- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )
Theorem	r2al 2561*	Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
|- ( A. x e. A A. y e. B ph <-> A. x A. y ( ( x e. A /\ y e. B ) -> ph ) )
Theorem	r2ex 2562*	Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)
|- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )
Theorem	2ralbida 2563*	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 24-Feb-2004.)
|- F/ x ph   &    |- F/ y ph   &    |- ( ( 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	2ralbidva 2564*	Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.)
|- ( ( 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	2rexbidva 2565*	Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004.)
|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) )
Theorem	2ralbidv 2566*	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 2567*	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 2568*	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 2569	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 2570*	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 2571	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 2572	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 2573	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 2574*	Not-free for restricted universal quantification where x and y are distinct. See nfraldya 2577 for a version with y and A distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by GG, 10-Jan-2024.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x A. y e. A ps )
Theorem	nfraldxy 2575*	Old name for nfraldw 2574. (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 2576*	Not-free for restricted existential quantification where x and y are distinct. See nfrexdya 2578 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 2577*	Not-free for restricted universal quantification where y and A are distinct. See nfraldxy 2575 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 2578*	Not-free for restricted existential quantification where y and A are distinct. See nfrexdxy 2576 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 2579*	Bound-variable hypothesis builder for restricted quantification. See nfralya 2582 for a version with y and A distinct instead of x and y. (Contributed by NM, 1-Sep-1999.) (Revised by GG, 10-Jan-2024.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x A. y e. A ph
Theorem	nfralxy 2580*	Old name for nfralw 2579. (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	nfrexw 2581*	Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2583 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 2582*	Not-free for restricted universal quantification where y and A are distinct. See nfralxy 2580 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 2583*	Not-free for restricted existential quantification where y and A are distinct. See nfrexw 2581 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 2584*	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 2585	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 2586*	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 2587	Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
|- ( A. x ph -> A. x e. A ph )
Theorem	rexex 2588	Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
|- ( E. x e. A ph -> E. x ph )
Theorem	rsp 2589	Restricted specialization. (Contributed by NM, 17-Oct-1996.)
|- ( A. x e. A ph -> ( x e. A -> ph ) )
Theorem	rspa 2590	Restricted specialization. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A. x e. A ph /\ x e. A ) -> ph )
Theorem	rspe 2591	Restricted specialization. (Contributed by NM, 12-Oct-1999.)
|- ( ( x e. A /\ ph ) -> E. x e. A ph )
Theorem	rsp2 2592	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 2593	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 2594	Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
|- A. x e. A ph   =>    |- ( x e. A -> ph )
Theorem	rgen 2595	Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
|- ( x e. A -> ph )   =>    |- A. x e. A ph
Theorem	rgen2a 2596*	Generalization rule for restricted quantification. Note that x and y are not required to be disjoint. This proof illustrates the use of dvelim 2071. Usage of rgen2 2628 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 2597	Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
|- ph   =>    |- A. x e. A ph
Theorem	rgen2w 2598	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 2599	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 2600	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