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	dvelimdc 2301	Deduction form of dvelimc 2302. (Contributed by Mario Carneiro, 8-Oct-2016.)
|- F/ x ph   &    |- F/ z ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/_ z B )   &    |- ( ph -> ( z = y -> A = B ) )   =>    |- ( ph -> ( -. A. x x = y -> F/_ x B ) )
Theorem	dvelimc 2302	Version of dvelim 1992 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
|- F/_ x A   &    |- F/_ z B   &    |- ( z = y -> A = B )   =>    |- ( -. A. x x = y -> F/_ x B )
Theorem	nfcvf 2303	If x and y are distinct, then x is not free in y. (Contributed by Mario Carneiro, 8-Oct-2016.)
|- ( -. A. x x = y -> F/_ x y )
Theorem	nfcvf2 2304	If x and y are distinct, then y is not free in x. (Contributed by Mario Carneiro, 5-Dec-2016.)
|- ( -. A. x x = y -> F/_ y x )
Theorem	cleqf 2305	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2239. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A = B <-> A. x ( x e. A <-> x e. B ) )
Theorem	abid2f 2306	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/_ x A   =>    |- { x | x e. A } = A
Theorem	sbabel 2307*	Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/_ x A   =>    |- ( [ y / x ] { z | ph } e. A <-> { z | [ y / x ] ph } e. A )
2.1.4  Negated equality and membership
2.1.4.1  Negated equality
Syntax	wne 2308	Extend wff notation to include inequality.
wff A =/= B
Definition	df-ne 2309	Define inequality. (Contributed by NM, 5-Aug-1993.)
|- ( A =/= B <-> -. A = B )
Theorem	neii 2310	Inference associated with df-ne 2309. (Contributed by BJ, 7-Jul-2018.)
|- A =/= B   =>    |- -. A = B
Theorem	neir 2311	Inference associated with df-ne 2309. (Contributed by BJ, 7-Jul-2018.)
|- -. A = B   =>    |- A =/= B
Theorem	nner 2312	Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
|- ( A = B -> -. A =/= B )
Theorem	nnedc 2313	Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
|- (DECID A = B -> ( -. A =/= B <-> A = B ) )
Theorem	dcned 2314	Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
|- ( ph -> DECID A = B )   =>    |- ( ph -> DECID A =/= B )
Theorem	neqned 2315	If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2329. One-way deduction form of df-ne 2309. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2358. (Revised by Wolf Lammen, 22-Nov-2019.)
|- ( ph -> -. A = B )   =>    |- ( ph -> A =/= B )
Theorem	neqne 2316	From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( -. A = B -> A =/= B )
Theorem	neirr 2317	No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
|- -. A =/= A
Theorem	eqneqall 2318	A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
|- ( A = B -> ( A =/= B -> ph ) )
Theorem	dcne 2319	Decidable equality expressed in terms of =/=. Basically the same as df-dc 820. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- (DECID A = B <-> ( A = B \/ A =/= B ) )
Theorem	nonconne 2320	Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
|- -. ( A = B /\ A =/= B )
Theorem	neeq1 2321	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
|- ( A = B -> ( A =/= C <-> B =/= C ) )
Theorem	neeq2 2322	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
|- ( A = B -> ( C =/= A <-> C =/= B ) )
Theorem	neeq1i 2323	Inference for inequality. (Contributed by NM, 29-Apr-2005.)
|- A = B   =>    |- ( A =/= C <-> B =/= C )
Theorem	neeq2i 2324	Inference for inequality. (Contributed by NM, 29-Apr-2005.)
|- A = B   =>    |- ( C =/= A <-> C =/= B )
Theorem	neeq12i 2325	Inference for inequality. (Contributed by NM, 24-Jul-2012.)
|- A = B   &    |- C = D   =>    |- ( A =/= C <-> B =/= D )
Theorem	neeq1d 2326	Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A =/= C <-> B =/= C ) )
Theorem	neeq2d 2327	Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C =/= A <-> C =/= B ) )
Theorem	neeq12d 2328	Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A =/= C <-> B =/= D ) )
Theorem	neneqd 2329	Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph -> A =/= B )   =>    |- ( ph -> -. A = B )
Theorem	neneq 2330	From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A =/= B -> -. A = B )
Theorem	eqnetri 2331	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- A = B   &    |- B =/= C   =>    |- A =/= C
Theorem	eqnetrd 2332	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- ( ph -> A = B )   &    |- ( ph -> B =/= C )   =>    |- ( ph -> A =/= C )
Theorem	eqnetrri 2333	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- A = B   &    |- A =/= C   =>    |- B =/= C
Theorem	eqnetrrd 2334	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- ( ph -> A = B )   &    |- ( ph -> A =/= C )   =>    |- ( ph -> B =/= C )
Theorem	neeqtri 2335	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- A =/= B   &    |- B = C   =>    |- A =/= C
Theorem	neeqtrd 2336	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- ( ph -> A =/= B )   &    |- ( ph -> B = C )   =>    |- ( ph -> A =/= C )
Theorem	neeqtrri 2337	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- A =/= B   &    |- C = B   =>    |- A =/= C
Theorem	neeqtrrd 2338	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- ( ph -> A =/= B )   &    |- ( ph -> C = B )   =>    |- ( ph -> A =/= C )
Theorem	eqnetrrid 2339	B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
|- B = A   &    |- ( ph -> B =/= C )   =>    |- ( ph -> A =/= C )
Theorem	3netr3d 2340	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
|- ( ph -> A =/= B )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> C =/= D )
Theorem	3netr4d 2341	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
|- ( ph -> A =/= B )   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> C =/= D )
Theorem	3netr3g 2342	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
|- ( ph -> A =/= B )   &    |- A = C   &    |- B = D   =>    |- ( ph -> C =/= D )
Theorem	3netr4g 2343	Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
|- ( ph -> A =/= B )   &    |- C = A   &    |- D = B   =>    |- ( ph -> C =/= D )
Theorem	necon3abii 2344	Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
|- ( A = B <-> ph )   =>    |- ( A =/= B <-> -. ph )
Theorem	necon3bbii 2345	Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
|- ( ph <-> A = B )   =>    |- ( -. ph <-> A =/= B )
Theorem	necon3bii 2346	Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
|- ( A = B <-> C = D )   =>    |- ( A =/= B <-> C =/= D )
Theorem	necon3abid 2347	Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
|- ( ph -> ( A = B <-> ps ) )   =>    |- ( ph -> ( A =/= B <-> -. ps ) )
Theorem	necon3bbid 2348	Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
|- ( ph -> ( ps <-> A = B ) )   =>    |- ( ph -> ( -. ps <-> A =/= B ) )
Theorem	necon3bid 2349	Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> ( A = B <-> C = D ) )   =>    |- ( ph -> ( A =/= B <-> C =/= D ) )
Theorem	necon3ad 2350	Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
|- ( ph -> ( ps -> A = B ) )   =>    |- ( ph -> ( A =/= B -> -. ps ) )
Theorem	necon3bd 2351	Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
|- ( ph -> ( A = B -> ps ) )   =>    |- ( ph -> ( -. ps -> A =/= B ) )
Theorem	necon3d 2352	Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
|- ( ph -> ( A = B -> C = D ) )   =>    |- ( ph -> ( C =/= D -> A =/= B ) )
Theorem	nesym 2353	Characterization of inequality in terms of reversed equality (see bicom 139). (Contributed by BJ, 7-Jul-2018.)
|- ( A =/= B <-> -. B = A )
Theorem	nesymi 2354	Inference associated with nesym 2353. (Contributed by BJ, 7-Jul-2018.)
|- A =/= B   =>    |- -. B = A
Theorem	nesymir 2355	Inference associated with nesym 2353. (Contributed by BJ, 7-Jul-2018.)
|- -. A = B   =>    |- B =/= A
Theorem	necon3i 2356	Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)
|- ( A = B -> C = D )   =>    |- ( C =/= D -> A =/= B )
Theorem	necon3ai 2357	Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
|- ( ph -> A = B )   =>    |- ( A =/= B -> -. ph )
Theorem	necon3bi 2358	Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
|- ( A = B -> ph )   =>    |- ( -. ph -> A =/= B )
Theorem	necon1aidc 2359	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
|- (DECID ph -> ( -. ph -> A = B ) )   =>    |- (DECID ph -> ( A =/= B -> ph ) )
Theorem	necon1bidc 2360	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
|- (DECID A = B -> ( A =/= B -> ph ) )   =>    |- (DECID A = B -> ( -. ph -> A = B ) )
Theorem	necon1idc 2361	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- ( A =/= B -> C = D )   =>    |- (DECID A = B -> ( C =/= D -> A = B ) )
Theorem	necon2ai 2362	Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
|- ( A = B -> -. ph )   =>    |- ( ph -> A =/= B )
Theorem	necon2bi 2363	Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
|- ( ph -> A =/= B )   =>    |- ( A = B -> -. ph )
Theorem	necon2i 2364	Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
|- ( A = B -> C =/= D )   =>    |- ( C = D -> A =/= B )
Theorem	necon2ad 2365	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 2366	Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
|- ( ph -> ( ps -> A =/= B ) )   =>    |- ( ph -> ( A = B -> -. ps ) )
Theorem	necon2d 2367	Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
|- ( ph -> ( A = B -> C =/= D ) )   =>    |- ( ph -> ( C = D -> A =/= B ) )
Theorem	necon1abiidc 2368	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- (DECID ph -> ( -. ph <-> A = B ) )   =>    |- (DECID ph -> ( A =/= B <-> ph ) )
Theorem	necon1bbiidc 2369	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 2370	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 2371	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 2372	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- (DECID ph -> ( A = B <-> -. ph ) )   =>    |- (DECID ph -> ( ph <-> A =/= B ) )
Theorem	necon2bbiidc 2373	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 2374	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 2375	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 2376	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 2377	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 2378	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 2379	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 2380	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 2381	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 2382	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 2383	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 2384	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 2385	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 2386	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 2387	If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2329. One-way deduction form of df-ne 2309. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> -. A = B )   =>    |- ( ph -> A =/= B )
Theorem	nebidc 2388	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 2389	Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( ( A = B /\ A =/= C ) -> B =/= C )
Theorem	pm13.181 2390	Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( ( A = B /\ B =/= C ) -> A =/= C )
Theorem	pm2.21ddne 2391	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 2392	Commutation of inequality. (Contributed by NM, 14-May-1999.)
|- ( A =/= B <-> B =/= A )
Theorem	necomi 2393	Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
|- A =/= B   =>    |- B =/= A
Theorem	necomd 2394	Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
|- ( ph -> A =/= B )   =>    |- ( ph -> B =/= A )
Theorem	neanior 2395	A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
|- ( ( A =/= B /\ C =/= D ) <-> -. ( A = B \/ C = D ) )
Theorem	ne3anior 2396	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 2397	An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
|- A =/= B   &    |- ( ph <-> A = B )   =>    |- -. ph
Theorem	nelne1 2398	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 2399	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 2400	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 ) )