P. 23 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2201-2300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cleqf 2201	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2137. (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 2202	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 2203*	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
Syntax	wne 2204	Extend wff notation to include inequality.
wff A =/= B
Syntax	wnel 2205	Extend wff notation to include negated membership.
wff A e/ B
Definition	df-ne 2206	Define inequality. (Contributed by NM, 5-Aug-1993.)
|- ( A =/= B <-> -. A = B )
Definition	df-nel 2207	Define negated membership. (Contributed by NM, 7-Aug-1994.)
|- ( A e/ B <-> -. A e. B )
2.1.4.1  Negated equality
Theorem	neii 2208	Inference associated with df-ne 2206. (Contributed by BJ, 7-Jul-2018.)
|- A =/= B   =>    |- -. A = B
Theorem	neir 2209	Inference associated with df-ne 2206. (Contributed by BJ, 7-Jul-2018.)
|- -. A = B   =>    |- A =/= B
Theorem	nner 2210	Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
|- ( A = B -> -. A =/= B )
Theorem	nnedc 2211	Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
|- (DECID A = B -> ( -. A =/= B <-> A = B ) )
Theorem	dcned 2212	Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
|- ( ph -> DECID A = B )   =>    |- ( ph -> DECID A =/= B )
Theorem	neqned 2213	If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2226. One-way deduction form of df-ne 2206. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2255. (Revised by Wolf Lammen, 22-Nov-2019.)
|- ( ph -> -. A = B )   =>    |- ( ph -> A =/= B )
Theorem	neqne 2214	From non equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( -. A = B -> A =/= B )
Theorem	neirr 2215	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	dcne 2216	Decidable equality expressed in terms of =/=. Basically the same as df-dc 743. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- (DECID A = B <-> ( A = B \/ A =/= B ) )
Theorem	nonconne 2217	Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
|- -. ( A = B /\ A =/= B )
Theorem	neeq1 2218	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
|- ( A = B -> ( A =/= C <-> B =/= C ) )
Theorem	neeq2 2219	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
|- ( A = B -> ( C =/= A <-> C =/= B ) )
Theorem	neeq1i 2220	Inference for inequality. (Contributed by NM, 29-Apr-2005.)
|- A = B   =>    |- ( A =/= C <-> B =/= C )
Theorem	neeq2i 2221	Inference for inequality. (Contributed by NM, 29-Apr-2005.)
|- A = B   =>    |- ( C =/= A <-> C =/= B )
Theorem	neeq12i 2222	Inference for inequality. (Contributed by NM, 24-Jul-2012.)
|- A = B   &    |- C = D   =>    |- ( A =/= C <-> B =/= D )
Theorem	neeq1d 2223	Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A =/= C <-> B =/= C ) )
Theorem	neeq2d 2224	Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C =/= A <-> C =/= B ) )
Theorem	neeq12d 2225	Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A =/= C <-> B =/= D ) )
Theorem	neneqd 2226	Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph -> A =/= B )   =>    |- ( ph -> -. A = B )
Theorem	neneq 2227	From inequality to non equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A =/= B -> -. A = B )
Theorem	eqnetri 2228	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- A = B   &    |- B =/= C   =>    |- A =/= C
Theorem	eqnetrd 2229	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- ( ph -> A = B )   &    |- ( ph -> B =/= C )   =>    |- ( ph -> A =/= C )
Theorem	eqnetrri 2230	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- A = B   &    |- A =/= C   =>    |- B =/= C
Theorem	eqnetrrd 2231	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- ( ph -> A = B )   &    |- ( ph -> A =/= C )   =>    |- ( ph -> B =/= C )
Theorem	neeqtri 2232	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- A =/= B   &    |- B = C   =>    |- A =/= C
Theorem	neeqtrd 2233	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- ( ph -> A =/= B )   &    |- ( ph -> B = C )   =>    |- ( ph -> A =/= C )
Theorem	neeqtrri 2234	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- A =/= B   &    |- C = B   =>    |- A =/= C
Theorem	neeqtrrd 2235	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- ( ph -> A =/= B )   &    |- ( ph -> C = B )   =>    |- ( ph -> A =/= C )
Theorem	syl5eqner 2236	B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
|- B = A   &    |- ( ph -> B =/= C )   =>    |- ( ph -> A =/= C )
Theorem	3netr3d 2237	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 2238	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 2239	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 2240	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 2241	Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
|- ( A = B <-> ph )   =>    |- ( A =/= B <-> -. ph )
Theorem	necon3bbii 2242	Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
|- ( ph <-> A = B )   =>    |- ( -. ph <-> A =/= B )
Theorem	necon3bii 2243	Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
|- ( A = B <-> C = D )   =>    |- ( A =/= B <-> C =/= D )
Theorem	necon3abid 2244	Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
|- ( ph -> ( A = B <-> ps ) )   =>    |- ( ph -> ( A =/= B <-> -. ps ) )
Theorem	necon3bbid 2245	Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
|- ( ph -> ( ps <-> A = B ) )   =>    |- ( ph -> ( -. ps <-> A =/= B ) )
Theorem	necon3bid 2246	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 2247	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 2248	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 2249	Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
|- ( ph -> ( A = B -> C = D ) )   =>    |- ( ph -> ( C =/= D -> A =/= B ) )
Theorem	nesym 2250	Characterization of inequality in terms of reversed equality (see bicom 128). (Contributed by BJ, 7-Jul-2018.)
|- ( A =/= B <-> -. B = A )
Theorem	nesymi 2251	Inference associated with nesym 2250. (Contributed by BJ, 7-Jul-2018.)
|- A =/= B   =>    |- -. B = A
Theorem	nesymir 2252	Inference associated with nesym 2250. (Contributed by BJ, 7-Jul-2018.)
|- -. A = B   =>    |- B =/= A
Theorem	necon3i 2253	Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)
|- ( A = B -> C = D )   =>    |- ( C =/= D -> A =/= B )
Theorem	necon3ai 2254	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 2255	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 2256	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
|- (DECID ph -> ( -. ph -> A = B ) )   =>    |- (DECID ph -> ( A =/= B -> ph ) )
Theorem	necon1bidc 2257	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 2258	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- ( A =/= B -> C = D )   =>    |- (DECID A = B -> ( C =/= D -> A = B ) )
Theorem	necon2ai 2259	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 2260	Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
|- ( ph -> A =/= B )   =>    |- ( A = B -> -. ph )
Theorem	necon2i 2261	Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
|- ( A = B -> C =/= D )   =>    |- ( C = D -> A =/= B )
Theorem	necon2ad 2262	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 2263	Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
|- ( ph -> ( ps -> A =/= B ) )   =>    |- ( ph -> ( A = B -> -. ps ) )
Theorem	necon2d 2264	Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
|- ( ph -> ( A = B -> C =/= D ) )   =>    |- ( ph -> ( C = D -> A =/= B ) )
Theorem	necon1abiidc 2265	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- (DECID ph -> ( -. ph <-> A = B ) )   =>    |- (DECID ph -> ( A =/= B <-> ph ) )
Theorem	necon1bbiidc 2266	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 2267	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 2268	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 2269	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
|- (DECID ph -> ( A = B <-> -. ph ) )   =>    |- (DECID ph -> ( ph <-> A =/= B ) )
Theorem	necon2bbii 2270	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 2271	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 2272	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 2273	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 2274	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 2275	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 2276	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 2277	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 2278	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 2279	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 2280	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 2281	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 2282	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 2283	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 2284	If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2226. One-way deduction form of df-ne 2206. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> -. A = B )   =>    |- ( ph -> A =/= B )
Theorem	nebidc 2285	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 2286	Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( ( A = B /\ A =/= C ) -> B =/= C )
Theorem	pm13.181 2287	Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( ( A = B /\ B =/= C ) -> A =/= C )
Theorem	pm2.21ddne 2288	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 2289	Commutation of inequality. (Contributed by NM, 14-May-1999.)
|- ( A =/= B <-> B =/= A )
Theorem	necomi 2290	Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
|- A =/= B   =>    |- B =/= A
Theorem	necomd 2291	Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
|- ( ph -> A =/= B )   =>    |- ( ph -> B =/= A )
Theorem	neanior 2292	A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
|- ( ( A =/= B /\ C =/= D ) <-> -. ( A = B \/ C = D ) )
Theorem	ne3anior 2293	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 2294	An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
|- A =/= B   &    |- ( ph <-> A = B )   =>    |- -. ph
Theorem	nelne1 2295	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 2296	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 2297	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 2298	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
Theorem	neli 2299	Inference associated with df-nel 2207. (Contributed by BJ, 7-Jul-2018.)
|- A e/ B   =>    |- -. A e. B
Theorem	nelir 2300	Inference associated with df-nel 2207. (Contributed by BJ, 7-Jul-2018.)
|- -. A e. B   =>    |- A e/ B