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	nfne 2401	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 2402	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 2403	Extend wff notation to include negated membership.
wff A e/ B
Definition	df-nel 2404	Define negated membership. (Contributed by NM, 7-Aug-1994.)
|- ( A e/ B <-> -. A e. B )
Theorem	neli 2405	Inference associated with df-nel 2404. (Contributed by BJ, 7-Jul-2018.)
|- A e/ B   =>    |- -. A e. B
Theorem	nelir 2406	Inference associated with df-nel 2404. (Contributed by BJ, 7-Jul-2018.)
|- -. A e. B   =>    |- A e/ B
Theorem	neleq1 2407	Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
|- ( A = B -> ( A e/ C <-> B e/ C ) )
Theorem	neleq2 2408	Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
|- ( A = B -> ( C e/ A <-> C e/ B ) )
Theorem	neleq12d 2409	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 2410	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 2411	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 2412	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 2413	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 2414	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 2415	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 2416	Extend wff notation to include restricted universal quantification.
wff A. x e. A ph
Syntax	wrex 2417	Extend wff notation to include restricted existential quantification.
wff E. x e. A ph
Syntax	wreu 2418	Extend wff notation to include restricted existential uniqueness.
wff E! x e. A ph
Syntax	wrmo 2419	Extend wff notation to include restricted "at most one."
wff E* x e. A ph
Syntax	crab 2420	Extend class notation to include the restricted class abstraction (class builder).
class { x e. A | ph }
Definition	df-ral 2421	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 2422	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 2423	Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
|- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
Definition	df-rmo 2424	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 2425	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 2426	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 2427	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	dfrex2dc 2428	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 2429	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 2430	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 2431	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 2432	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 2433*	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 2434*	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 2435	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 2436	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 2437*	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 2438*	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 2439*	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 2440*	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	ralbii 2441	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 2442	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 2443	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 2444	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 2445	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 2446	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 2447	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 2448	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 2449	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 2450	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 2451*	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 2452*	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 2453*	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 2454*	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 2455*	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 2456*	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 2457*	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 2458*	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 2459*	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 2460*	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 2461*	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 2462	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 2463*	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 2464	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 2465	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 2466	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	nfraldxy 2467*	Not-free for restricted universal quantification where x and y are distinct. See nfraldya 2469 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 29-May-2018.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x A. y e. A ps )
Theorem	nfrexdxy 2468*	Not-free for restricted existential quantification where x and y are distinct. See nfrexdya 2470 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 2469*	Not-free for restricted universal quantification where y and A are distinct. See nfraldxy 2467 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 2470*	Not-free for restricted existential quantification where y and A are distinct. See nfrexdxy 2468 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	nfralxy 2471*	Not-free for restricted universal quantification where x and y are distinct. See nfralya 2473 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x A. y e. A ph
Theorem	nfrexxy 2472*	Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2474 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 2473*	Not-free for restricted universal quantification where y and A are distinct. See nfralxy 2471 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 2474*	Not-free for restricted existential quantification where y and A are distinct. See nfrexxy 2472 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 2475*	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 2476	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 2477*	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 2478	Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
|- ( A. x ph -> A. x e. A ph )
Theorem	rexex 2479	Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
|- ( E. x e. A ph -> E. x ph )
Theorem	rsp 2480	Restricted specialization. (Contributed by NM, 17-Oct-1996.)
|- ( A. x e. A ph -> ( x e. A -> ph ) )
Theorem	rspe 2481	Restricted specialization. (Contributed by NM, 12-Oct-1999.)
|- ( ( x e. A /\ ph ) -> E. x e. A ph )
Theorem	rsp2 2482	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 2483	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 2484	Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
|- A. x e. A ph   =>    |- ( x e. A -> ph )
Theorem	rgen 2485	Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
|- ( x e. A -> ph )   =>    |- A. x e. A ph
Theorem	rgen2a 2486*	Generalization rule for restricted quantification. Note that x and y needn't be distinct (and illustrates the use of dvelimor 1993). (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.)
|- ( ( x e. A /\ y e. A ) -> ph )   =>    |- A. x e. A A. y e. A ph
Theorem	rgenw 2487	Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
|- ph   =>    |- A. x e. A ph
Theorem	rgen2w 2488	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 2489	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 2490	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 2491	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 2492	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 2493	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 2494	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 2495	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 2496	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 2497	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 2498	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 2499*	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 2500*	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 ) )