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Theorem List for Intuitionistic Logic Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfeq 2201 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  A  =  B
 
Theoremnfel 2202 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  A  e.  B
 
Theoremnfeq1 2203* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x  A  =  B
 
Theoremnfel1 2204* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x  A  e.  B
 
Theoremnfeq2 2205* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x B   =>    |- 
 F/ x  A  =  B
 
Theoremnfel2 2206* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x B   =>    |- 
 F/ x  A  e.  B
 
Theoremnfcrd 2207* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/ x  y  e.  A )
 
Theoremnfeqd 2208 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/ x  A  =  B )
 
Theoremnfeld 2209 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/ x  A  e.  B )
 
Theoremdrnfc1 2210 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
 |-  ( A. x  x  =  y  ->  A  =  B )   =>    |-  ( A. x  x  =  y  ->  ( F/_ x A  <->  F/_ y B ) )
 
Theoremdrnfc2 2211 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
 |-  ( A. x  x  =  y  ->  A  =  B )   =>    |-  ( A. x  x  =  y  ->  ( F/_ z A  <->  F/_ z B ) )
 
Theoremnfabd 2212 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/_ x { y  |  ps } )
 
Theoremdvelimdc 2213 Deduction form of dvelimc 2214. (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 ) )
 
Theoremdvelimc 2214 Version of dvelim 1909 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 )
 
Theoremnfcvf 2215 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 )
 
Theoremnfcvf2 2216 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 )
 
Theoremcleqf 2217 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2153. (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 ) )
 
Theoremabid2f 2218 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
 
Theoremsbabel 2219* 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
 
Syntaxwne 2220 Extend wff notation to include inequality.
 wff  A  =/=  B
 
Definitiondf-ne 2221 Define inequality. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =/=  B  <->  -.  A  =  B )
 
Theoremneii 2222 Inference associated with df-ne 2221. (Contributed by BJ, 7-Jul-2018.)
 |-  A  =/=  B   =>    |-  -.  A  =  B
 
Theoremneir 2223 Inference associated with df-ne 2221. (Contributed by BJ, 7-Jul-2018.)
 |- 
 -.  A  =  B   =>    |-  A  =/=  B
 
Theoremnner 2224 Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
 |-  ( A  =  B  ->  -.  A  =/=  B )
 
Theoremnnedc 2225 Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
 |-  (DECID  A  =  B  ->  ( -.  A  =/=  B  <->  A  =  B ) )
 
Theoremdcned 2226 Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
 |-  ( ph  -> DECID  A  =  B )   =>    |-  ( ph  -> DECID  A  =/=  B )
 
Theoremneqned 2227 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2241. One-way deduction form of df-ne 2221. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2270. (Revised by Wolf Lammen, 22-Nov-2019.)
 |-  ( ph  ->  -.  A  =  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremneqne 2228 From non equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( -.  A  =  B  ->  A  =/=  B )
 
Theoremneirr 2229 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
 |- 
 -.  A  =/=  A
 
Theoremeqneqall 2230 A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  ( A  =  B  ->  ( A  =/=  B  -> 
 ph ) )
 
Theoremdcne 2231 Decidable equality expressed in terms of  =/=. Basically the same as df-dc 754. (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  (DECID  A  =  B  <->  ( A  =  B  \/  A  =/=  B ) )
 
Theoremnonconne 2232 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
 |- 
 -.  ( A  =  B  /\  A  =/=  B )
 
Theoremneeq1 2233 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
 |-  ( A  =  B  ->  ( A  =/=  C  <->  B  =/=  C ) )
 
Theoremneeq2 2234 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
 |-  ( A  =  B  ->  ( C  =/=  A  <->  C  =/=  B ) )
 
Theoremneeq1i 2235 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
 |-  A  =  B   =>    |-  ( A  =/=  C  <->  B  =/=  C )
 
Theoremneeq2i 2236 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
 |-  A  =  B   =>    |-  ( C  =/=  A  <->  C  =/=  B )
 
Theoremneeq12i 2237 Inference for inequality. (Contributed by NM, 24-Jul-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  =/=  C  <->  B  =/=  D )
 
Theoremneeq1d 2238 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  C ) )
 
Theoremneeq2d 2239 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  =/=  A  <->  C  =/=  B ) )
 
Theoremneeq12d 2240 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  D ) )
 
Theoremneneqd 2241 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  -.  A  =  B )
 
Theoremneneq 2242 From inequality to non equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  =/=  B  ->  -.  A  =  B )
 
Theoremeqnetri 2243 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =  B   &    |-  B  =/=  C   =>    |-  A  =/=  C
 
Theoremeqnetrd 2244 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremeqnetrri 2245 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =  B   &    |-  A  =/=  C   =>    |-  B  =/=  C
 
Theoremeqnetrrd 2246 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  B  =/=  C )
 
Theoremneeqtri 2247 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =/=  B   &    |-  B  =  C   =>    |-  A  =/=  C
 
Theoremneeqtrd 2248 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremneeqtrri 2249 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =/=  B   &    |-  C  =  B   =>    |-  A  =/=  C
 
Theoremneeqtrrd 2250 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremsyl5eqner 2251 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
 |-  B  =  A   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theorem3netr3d 2252 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 )
 
Theorem3netr4d 2253 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 )
 
Theorem3netr3g 2254 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 )
 
Theorem3netr4g 2255 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 )
 
Theoremnecon3abii 2256 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
 |-  ( A  =  B  <->  ph )   =>    |-  ( A  =/=  B  <->  -.  ph )
 
Theoremnecon3bbii 2257 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
 |-  ( ph  <->  A  =  B )   =>    |-  ( -.  ph  <->  A  =/=  B )
 
Theoremnecon3bii 2258 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
 |-  ( A  =  B  <->  C  =  D )   =>    |-  ( A  =/=  B  <->  C  =/=  D )
 
Theoremnecon3abid 2259 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
 |-  ( ph  ->  ( A  =  B  <->  ps ) )   =>    |-  ( ph  ->  ( A  =/=  B  <->  -.  ps ) )
 
Theoremnecon3bbid 2260 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
 |-  ( ph  ->  ( ps 
 <->  A  =  B ) )   =>    |-  ( ph  ->  ( -.  ps  <->  A  =/=  B ) )
 
Theoremnecon3bid 2261 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 ) )
 
Theoremnecon3ad 2262 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 ) )
 
Theoremnecon3bd 2263 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 ) )
 
Theoremnecon3d 2264 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
 |-  ( ph  ->  ( A  =  B  ->  C  =  D ) )   =>    |-  ( ph  ->  ( C  =/=  D  ->  A  =/=  B ) )
 
Theoremnesym 2265 Characterization of inequality in terms of reversed equality (see bicom 132). (Contributed by BJ, 7-Jul-2018.)
 |-  ( A  =/=  B  <->  -.  B  =  A )
 
Theoremnesymi 2266 Inference associated with nesym 2265. (Contributed by BJ, 7-Jul-2018.)
 |-  A  =/=  B   =>    |-  -.  B  =  A
 
Theoremnesymir 2267 Inference associated with nesym 2265. (Contributed by BJ, 7-Jul-2018.)
 |- 
 -.  A  =  B   =>    |-  B  =/=  A
 
Theoremnecon3i 2268 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)
 |-  ( A  =  B  ->  C  =  D )   =>    |-  ( C  =/=  D  ->  A  =/=  B )
 
Theoremnecon3ai 2269 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( A  =/=  B  ->  -.  ph )
 
Theoremnecon3bi 2270 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
 |-  ( A  =  B  -> 
 ph )   =>    |-  ( -.  ph  ->  A  =/=  B )
 
Theoremnecon1aidc 2271 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
 |-  (DECID 
 ph  ->  ( -.  ph  ->  A  =  B ) )   =>    |-  (DECID 
 ph  ->  ( A  =/=  B 
 ->  ph ) )
 
Theoremnecon1bidc 2272 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
 |-  (DECID  A  =  B  ->  ( A  =/=  B  ->  ph ) )   =>    |-  (DECID  A  =  B  ->  ( -.  ph  ->  A  =  B ) )
 
Theoremnecon1idc 2273 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  ( A  =/=  B  ->  C  =  D )   =>    |-  (DECID  A  =  B  ->  ( C  =/=  D  ->  A  =  B ) )
 
Theoremnecon2ai 2274 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
 |-  ( A  =  B  ->  -.  ph )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremnecon2bi 2275 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
 |-  ( ph  ->  A  =/=  B )   =>    |-  ( A  =  B  ->  -.  ph )
 
Theoremnecon2i 2276 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
 |-  ( A  =  B  ->  C  =/=  D )   =>    |-  ( C  =  D  ->  A  =/=  B )
 
Theoremnecon2ad 2277 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 ) )
 
Theoremnecon2bd 2278 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
 |-  ( ph  ->  ( ps  ->  A  =/=  B ) )   =>    |-  ( ph  ->  ( A  =  B  ->  -. 
 ps ) )
 
Theoremnecon2d 2279 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
 |-  ( ph  ->  ( A  =  B  ->  C  =/=  D ) )   =>    |-  ( ph  ->  ( C  =  D  ->  A  =/=  B ) )
 
Theoremnecon1abiidc 2280 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  (DECID 
 ph  ->  ( -.  ph  <->  A  =  B ) )   =>    |-  (DECID 
 ph  ->  ( A  =/=  B  <->  ph ) )
 
Theoremnecon1bbiidc 2281 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  (DECID  A  =  B  ->  ( A  =/=  B  <->  ph ) )   =>    |-  (DECID  A  =  B  ->  ( -.  ph  <->  A  =  B ) )
 
Theoremnecon1abiddc 2282 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  <->  A  =  B ) ) )   =>    |-  ( ph  ->  (DECID  ps 
 ->  ( A  =/=  B  <->  ps ) ) )
 
Theoremnecon1bbiddc 2283 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 ) ) )
 
Theoremnecon2abiidc 2284 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  (DECID 
 ph  ->  ( A  =  B 
 <->  -.  ph ) )   =>    |-  (DECID 
 ph  ->  ( ph  <->  A  =/=  B ) )
 
Theoremnecon2bbii 2285 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  (DECID  A  =  B  ->  (
 ph 
 <->  A  =/=  B ) )   =>    |-  (DECID  A  =  B  ->  ( A  =  B  <->  -.  ph ) )
 
Theoremnecon2abiddc 2286 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  ( ph  ->  (DECID  ps  ->  ( A  =  B  <->  -. 
 ps ) ) )   =>    |-  ( ph  ->  (DECID  ps  ->  ( ps  <->  A  =/=  B ) ) )
 
Theoremnecon2bbiddc 2287 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 ) ) )
 
Theoremnecon4aidc 2288 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  (DECID  A  =  B  ->  ( A  =/=  B  ->  -.  ph ) )   =>    |-  (DECID  A  =  B  ->  (
 ph  ->  A  =  B ) )
 
Theoremnecon4idc 2289 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 ) )
 
Theoremnecon4addc 2290 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 ) ) )
 
Theoremnecon4bddc 2291 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
 |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  A  =/=  B ) ) )   =>    |-  ( ph  ->  (DECID  ps  ->  ( A  =  B  ->  ps ) ) )
 
Theoremnecon4ddc 2292 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 ) ) )
 
Theoremnecon4abiddc 2293 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 ) ) ) )
 
Theoremnecon4bbiddc 2294 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 ) ) ) )
 
Theoremnecon4biddc 2295 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 ) ) ) )
 
Theoremnecon1addc 2296 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
 |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  A  =  B ) ) )   =>    |-  ( ph  ->  (DECID  ps  ->  ( A  =/=  B  ->  ps ) ) )
 
Theoremnecon1bddc 2297 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 ) ) )
 
Theoremnecon1ddc 2298 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 ) ) )
 
Theoremneneqad 2299 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2241. One-way deduction form of df-ne 2221. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  -.  A  =  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremnebidc 2300 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 ) ) ) )
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