P. 24 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2301-2400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	necon2i 2301	Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
|- ( A = B -> C =/= D )   =>    |- ( C = D -> A =/= B )
Theorem	necon2ad 2302	Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
|- ( ph -> ( A = B -> -. ps ) )   =>    |- ( ph -> ( ps -> A =/= B ) )
Theorem	necon2bd 2303	Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
|- ( ph -> ( ps -> A =/= B ) )   =>    |- ( ph -> ( A = B -> -. ps ) )
Theorem	necon2d 2304	Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
|- ( ph -> ( A = B -> C =/= D ) )   =>    |- ( ph -> ( C = D -> A =/= B ) )
Theorem	necon1abiidc 2305	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- (DECID ph -> ( -. ph <-> A = B ) )   =>    |- (DECID ph -> ( A =/= B <-> ph ) )
Theorem	necon1bbiidc 2306	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- (DECID A = B -> ( A =/= B <-> ph ) )   =>    |- (DECID A = B -> ( -. ph <-> A = B ) )
Theorem	necon1abiddc 2307	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- ( ph -> (DECID ps -> ( -. ps <-> A = B ) ) )   =>    |- ( ph -> (DECID ps -> ( A =/= B <-> ps ) ) )
Theorem	necon1bbiddc 2308	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- ( ph -> (DECID A = B -> ( A =/= B <-> ps ) ) )   =>    |- ( ph -> (DECID A = B -> ( -. ps <-> A = B ) ) )
Theorem	necon2abiidc 2309	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- (DECID ph -> ( A = B <-> -. ph ) )   =>    |- (DECID ph -> ( ph <-> A =/= B ) )
Theorem	necon2bbii 2310	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- (DECID A = B -> ( ph <-> A =/= B ) )   =>    |- (DECID A = B -> ( A = B <-> -. ph ) )
Theorem	necon2abiddc 2311	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- ( ph -> (DECID ps -> ( A = B <-> -. ps ) ) )   =>    |- ( ph -> (DECID ps -> ( ps <-> A =/= B ) ) )
Theorem	necon2bbiddc 2312	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- ( ph -> (DECID A = B -> ( ps <-> A =/= B ) ) )   =>    |- ( ph -> (DECID A = B -> ( A = B <-> -. ps ) ) )
Theorem	necon4aidc 2313	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- (DECID A = B -> ( A =/= B -> -. ph ) )   =>    |- (DECID A = B -> ( ph -> A = B ) )
Theorem	necon4idc 2314	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- (DECID A = B -> ( A =/= B -> C =/= D ) )   =>    |- (DECID A = B -> ( C = D -> A = B ) )
Theorem	necon4addc 2315	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
|- ( ph -> (DECID A = B -> ( A =/= B -> -. ps ) ) )   =>    |- ( ph -> (DECID A = B -> ( ps -> A = B ) ) )
Theorem	necon4bddc 2316	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
|- ( ph -> (DECID ps -> ( -. ps -> A =/= B ) ) )   =>    |- ( ph -> (DECID ps -> ( A = B -> ps ) ) )
Theorem	necon4ddc 2317	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
|- ( ph -> (DECID A = B -> ( A =/= B -> C =/= D ) ) )   =>    |- ( ph -> (DECID A = B -> ( C = D -> A = B ) ) )
Theorem	necon4abiddc 2318	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)
|- ( ph -> (DECID A = B -> (DECID ps -> ( A =/= B <-> -. ps ) ) ) )   =>    |- ( ph -> (DECID A = B -> (DECID ps -> ( A = B <-> ps ) ) ) )
Theorem	necon4bbiddc 2319	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
|- ( ph -> (DECID ps -> (DECID A = B -> ( -. ps <-> A =/= B ) ) ) )   =>    |- ( ph -> (DECID ps -> (DECID A = B -> ( ps <-> A = B ) ) ) )
Theorem	necon4biddc 2320	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
|- ( ph -> (DECID A = B -> (DECID C = D -> ( A =/= B <-> C =/= D ) ) ) )   =>    |- ( ph -> (DECID A = B -> (DECID C = D -> ( A = B <-> C = D ) ) ) )
Theorem	necon1addc 2321	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
|- ( ph -> (DECID ps -> ( -. ps -> A = B ) ) )   =>    |- ( ph -> (DECID ps -> ( A =/= B -> ps ) ) )
Theorem	necon1bddc 2322	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
|- ( ph -> (DECID A = B -> ( A =/= B -> ps ) ) )   =>    |- ( ph -> (DECID A = B -> ( -. ps -> A = B ) ) )
Theorem	necon1ddc 2323	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
|- ( ph -> (DECID A = B -> ( A =/= B -> C = D ) ) )   =>    |- ( ph -> (DECID A = B -> ( C =/= D -> A = B ) ) )
Theorem	neneqad 2324	If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2266. One-way deduction form of df-ne 2246. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> -. A = B )   =>    |- ( ph -> A =/= B )
Theorem	nebidc 2325	Contraposition law for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
|- (DECID A = B -> (DECID C = D -> ( ( A = B <-> C = D ) <-> ( A =/= B <-> C =/= D ) ) ) )
Theorem	pm13.18 2326	Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( ( A = B /\ A =/= C ) -> B =/= C )
Theorem	pm13.181 2327	Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( ( A = B /\ B =/= C ) -> A =/= C )
Theorem	pm2.21ddne 2328	A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A = B )   &    |- ( ph -> A =/= B )   =>    |- ( ph -> ps )
Theorem	necom 2329	Commutation of inequality. (Contributed by NM, 14-May-1999.)
|- ( A =/= B <-> B =/= A )
Theorem	necomi 2330	Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
|- A =/= B   =>    |- B =/= A
Theorem	necomd 2331	Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
|- ( ph -> A =/= B )   =>    |- ( ph -> B =/= A )
Theorem	neanior 2332	A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
|- ( ( A =/= B /\ C =/= D ) <-> -. ( A = B \/ C = D ) )
Theorem	ne3anior 2333	A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.)
|- ( ( A =/= B /\ C =/= D /\ E =/= F ) <-> -. ( A = B \/ C = D \/ E = F ) )
Theorem	nemtbir 2334	An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
|- A =/= B   &    |- ( ph <-> A = B )   =>    |- -. ph
Theorem	nelne1 2335	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 2336	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	nfne 2337	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 2338	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 2339	Extend wff notation to include negated membership.
wff A e/ B
Definition	df-nel 2340	Define negated membership. (Contributed by NM, 7-Aug-1994.)
|- ( A e/ B <-> -. A e. B )
Theorem	neli 2341	Inference associated with df-nel 2340. (Contributed by BJ, 7-Jul-2018.)
|- A e/ B   =>    |- -. A e. B
Theorem	nelir 2342	Inference associated with df-nel 2340. (Contributed by BJ, 7-Jul-2018.)
|- -. A e. B   =>    |- A e/ B
Theorem	neleq1 2343	Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
|- ( A = B -> ( A e/ C <-> B e/ C ) )
Theorem	neleq2 2344	Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
|- ( A = B -> ( C e/ A <-> C e/ B ) )
Theorem	neleq12d 2345	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 2346	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 2347	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 )
2.1.5  Restricted quantification
Syntax	wral 2348	Extend wff notation to include restricted universal quantification.
wff A. x e. A ph
Syntax	wrex 2349	Extend wff notation to include restricted existential quantification.
wff E. x e. A ph
Syntax	wreu 2350	Extend wff notation to include restricted existential uniqueness.
wff E! x e. A ph
Syntax	wrmo 2351	Extend wff notation to include restricted "at most one."
wff E* x e. A ph
Syntax	crab 2352	Extend class notation to include the restricted class abstraction (class builder).
class { x e. A | ph }
Definition	df-ral 2353	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 2354	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 2355	Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
|- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
Definition	df-rmo 2356	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 2357	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 2358	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 2359	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	ralexim 2360	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 2361	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 2362	Formula-building rule for restricted universal quantifier (deduction rule). (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 2363	Formula-building rule for restricted existential quantifier (deduction rule). (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 2364*	Formula-building rule for restricted universal quantifier (deduction rule). (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 2365*	Formula-building rule for restricted existential quantifier (deduction rule). (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 2366	Formula-building rule for restricted universal quantifier (deduction rule). (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 2367	Formula-building rule for restricted existential quantifier (deduction rule). (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 2368*	Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )
Theorem	rexbidv 2369*	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )
Theorem	ralbidv2 2370*	Formula-building rule for restricted universal quantifier (deduction rule). (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 2371*	Formula-building rule for restricted existential quantifier (deduction rule). (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 2372	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 2373	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 2374	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 2375	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 2376	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 2377	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 2378	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 2379	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 2380	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 2381	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 2382*	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 2383*	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 2384*	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 2385*	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 2386*	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 2387*	Formula-building rule for restricted universal quantifier (deduction rule). (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 2388*	Formula-building rule for restricted universal quantifiers (deduction rule). (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 2389*	Formula-building rule for restricted existential quantifiers (deduction rule). (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 2390*	Formula-building rule for restricted universal quantifiers (deduction rule). (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 2391*	Formula-building rule for restricted existential quantifiers (deduction rule). (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 2392*	Formula-building rule for restricted quantifiers (deduction rule). (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 2393	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 2394*	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 2395	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 2396	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 2397	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 2398*	Not-free for restricted universal quantification where x and y are distinct. See nfraldya 2400 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 2399*	Not-free for restricted existential quantification where x and y are distinct. See nfrexdya 2401 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 2400*	Not-free for restricted universal quantification where y and A are distinct. See nfraldxy 2398 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 )