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Theorem List for Intuitionistic Logic Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabbi 2201 Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)
 |-  ( A. x (
 ph 
 <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
 
Theoremabbi2i 2202* Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  A  <->  ph )   =>    |-  A  =  { x  |  ph }
 
Theoremabbii 2203 Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   =>    |- 
 { x  |  ph }  =  { x  |  ps }
 
Theoremabbid 2204 Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
 
Theoremabbidv 2205* Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
 
Theoremabbi2dv 2206* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
 |-  ( ph  ->  ( x  e.  A  <->  ps ) )   =>    |-  ( ph  ->  A  =  { x  |  ps } )
 
Theoremabbi1dv 2207* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
 |-  ( ph  ->  ( ps 
 <->  x  e.  A ) )   =>    |-  ( ph  ->  { x  |  ps }  =  A )
 
Theoremabid2 2208* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
 |- 
 { x  |  x  e.  A }  =  A
 
Theoremsb8ab 2209 Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
 |- 
 F/ y ph   =>    |- 
 { x  |  ph }  =  { y  |  [ y  /  x ] ph }
 
Theoremcbvab 2210 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  { x  |  ph
 }  =  { y  |  ps }
 
Theoremcbvabv 2211* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  { x  |  ph
 }  =  { y  |  ps }
 
Theoremclelab 2212* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
 |-  ( A  e.  { x  |  ph }  <->  E. x ( x  =  A  /\  ph )
 )
 
Theoremclabel 2213* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
 |-  ( { x  |  ph
 }  e.  A  <->  E. y ( y  e.  A  /\  A. x ( x  e.  y  <->  ph ) ) )
 
Theoremsbab 2214* The right-hand side of the second equality is a way of representing proper substitution of  y for  x into a class variable. (Contributed by NM, 14-Sep-2003.)
 |-  ( x  =  y 
 ->  A  =  { z  |  [ y  /  x ] z  e.  A } )
 
2.1.3  Class form not-free predicate
 
Syntaxwnfc 2215 Extend wff definition to include the not-free predicate for classes.
 wff  F/_ x A
 
Theoremnfcjust 2216* Justification theorem for df-nfc 2217. (Contributed by Mario Carneiro, 13-Oct-2016.)
 |-  ( A. y F/ x  y  e.  A  <->  A. z F/ x  z  e.  A )
 
Definitiondf-nfc 2217* Define the not-free predicate for classes. This is read " x is not free in  A". Not-free means that the value of  x cannot affect the value of  A, e.g., any occurrence of  x in  A is effectively bound by a quantifier or something that expands to one (such as "there exists at most one"). It is defined in terms of the not-free predicate df-nf 1395 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
 
Theoremnfci 2218* Deduce that a class  A does not have  x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x  y  e.  A   =>    |-  F/_ x A
 
Theoremnfcii 2219* Deduce that a class  A does not have  x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( y  e.  A  ->  A. x  y  e.  A )   =>    |-  F/_ x A
 
Theoremnfcr 2220* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( F/_ x A  ->  F/ x  y  e.  A )
 
Theoremnfcrii 2221* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   =>    |-  ( y  e.  A  ->  A. x  y  e.  A )
 
Theoremnfcri 2222* Consequence of the not-free predicate. (Note that unlike nfcr 2220, this does not require  y and  A to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   =>    |- 
 F/ x  y  e.  A
 
Theoremnfcd 2223* Deduce that a class  A does not have  x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x  y  e.  A )   =>    |-  ( ph  ->  F/_ x A )
 
Theoremnfceqi 2224 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  A  =  B   =>    |-  ( F/_ x A 
 <-> 
 F/_ x B )
 
Theoremnfcxfr 2225 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  A  =  B   &    |-  F/_ x B   =>    |-  F/_ x A
 
Theoremnfcxfrd 2226 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  A  =  B   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/_ x A )
 
Theoremnfceqdf 2227 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  (
 F/_ x A  <->  F/_ x B ) )
 
Theoremnfcv 2228* If  x is disjoint from  A, then  x is not free in  A. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A
 
Theoremnfcvd 2229* If  x is disjoint from  A, then  x is not free in  A. (Contributed by Mario Carneiro, 7-Oct-2016.)
 |-  ( ph  ->  F/_ x A )
 
Theoremnfab1 2230 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x { x  |  ph
 }
 
Theoremnfnfc1 2231  x is bound in  F/_ x A. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x F/_ x A
 
Theoremclelsb3f 2232 Substitution applied to an atomic wff (class version of elsb3 1900). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
 |-  F/_ y A   =>    |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
 
Theoremnfab 2233 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |-  F/_ x { y  | 
 ph }
 
Theoremnfaba1 2234 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x { y  | 
 A. x ph }
 
Theoremnfnfc 2235 Hypothesis builder for  F/_ y A. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   =>    |- 
 F/ x F/_ y A
 
Theoremnfeq 2236 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  A  =  B
 
Theoremnfel 2237 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  A  e.  B
 
Theoremnfeq1 2238* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x  A  =  B
 
Theoremnfel1 2239* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x  A  e.  B
 
Theoremnfeq2 2240* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x B   =>    |- 
 F/ x  A  =  B
 
Theoremnfel2 2241* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x B   =>    |- 
 F/ x  A  e.  B
 
Theoremnfcrd 2242* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/ x  y  e.  A )
 
Theoremnfeqd 2243 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 2244 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 2245 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 2246 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 2247 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 2248 Deduction form of dvelimc 2249. (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 2249 Version of dvelim 1941 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 2250 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 2251 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 2252 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2187. (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 2253 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 2254* 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 2255 Extend wff notation to include inequality.
 wff  A  =/=  B
 
Definitiondf-ne 2256 Define inequality. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =/=  B  <->  -.  A  =  B )
 
Theoremneii 2257 Inference associated with df-ne 2256. (Contributed by BJ, 7-Jul-2018.)
 |-  A  =/=  B   =>    |-  -.  A  =  B
 
Theoremneir 2258 Inference associated with df-ne 2256. (Contributed by BJ, 7-Jul-2018.)
 |- 
 -.  A  =  B   =>    |-  A  =/=  B
 
Theoremnner 2259 Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
 |-  ( A  =  B  ->  -.  A  =/=  B )
 
Theoremnnedc 2260 Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
 |-  (DECID  A  =  B  ->  ( -.  A  =/=  B  <->  A  =  B ) )
 
Theoremdcned 2261 Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
 |-  ( ph  -> DECID  A  =  B )   =>    |-  ( ph  -> DECID  A  =/=  B )
 
Theoremneqned 2262 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2276. One-way deduction form of df-ne 2256. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2305. (Revised by Wolf Lammen, 22-Nov-2019.)
 |-  ( ph  ->  -.  A  =  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremneqne 2263 From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( -.  A  =  B  ->  A  =/=  B )
 
Theoremneirr 2264 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 2265 A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  ( A  =  B  ->  ( A  =/=  B  -> 
 ph ) )
 
Theoremdcne 2266 Decidable equality expressed in terms of  =/=. Basically the same as df-dc 781. (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  (DECID  A  =  B  <->  ( A  =  B  \/  A  =/=  B ) )
 
Theoremnonconne 2267 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
 |- 
 -.  ( A  =  B  /\  A  =/=  B )
 
Theoremneeq1 2268 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
 |-  ( A  =  B  ->  ( A  =/=  C  <->  B  =/=  C ) )
 
Theoremneeq2 2269 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
 |-  ( A  =  B  ->  ( C  =/=  A  <->  C  =/=  B ) )
 
Theoremneeq1i 2270 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
 |-  A  =  B   =>    |-  ( A  =/=  C  <->  B  =/=  C )
 
Theoremneeq2i 2271 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
 |-  A  =  B   =>    |-  ( C  =/=  A  <->  C  =/=  B )
 
Theoremneeq12i 2272 Inference for inequality. (Contributed by NM, 24-Jul-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  =/=  C  <->  B  =/=  D )
 
Theoremneeq1d 2273 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  C ) )
 
Theoremneeq2d 2274 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  =/=  A  <->  C  =/=  B ) )
 
Theoremneeq12d 2275 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  D ) )
 
Theoremneneqd 2276 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  -.  A  =  B )
 
Theoremneneq 2277 From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  =/=  B  ->  -.  A  =  B )
 
Theoremeqnetri 2278 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =  B   &    |-  B  =/=  C   =>    |-  A  =/=  C
 
Theoremeqnetrd 2279 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremeqnetrri 2280 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =  B   &    |-  A  =/=  C   =>    |-  B  =/=  C
 
Theoremeqnetrrd 2281 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  B  =/=  C )
 
Theoremneeqtri 2282 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =/=  B   &    |-  B  =  C   =>    |-  A  =/=  C
 
Theoremneeqtrd 2283 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremneeqtrri 2284 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =/=  B   &    |-  C  =  B   =>    |-  A  =/=  C
 
Theoremneeqtrrd 2285 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremsyl5eqner 2286 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
 |-  B  =  A   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theorem3netr3d 2287 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 2288 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 2289 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 2290 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 2291 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
 |-  ( A  =  B  <->  ph )   =>    |-  ( A  =/=  B  <->  -.  ph )
 
Theoremnecon3bbii 2292 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
 |-  ( ph  <->  A  =  B )   =>    |-  ( -.  ph  <->  A  =/=  B )
 
Theoremnecon3bii 2293 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
 |-  ( A  =  B  <->  C  =  D )   =>    |-  ( A  =/=  B  <->  C  =/=  D )
 
Theoremnecon3abid 2294 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
 |-  ( ph  ->  ( A  =  B  <->  ps ) )   =>    |-  ( ph  ->  ( A  =/=  B  <->  -.  ps ) )
 
Theoremnecon3bbid 2295 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
 |-  ( ph  ->  ( ps 
 <->  A  =  B ) )   =>    |-  ( ph  ->  ( -.  ps  <->  A  =/=  B ) )
 
Theoremnecon3bid 2296 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 2297 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 2298 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 2299 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
 |-  ( ph  ->  ( A  =  B  ->  C  =  D ) )   =>    |-  ( ph  ->  ( C  =/=  D  ->  A  =/=  B ) )
 
Theoremnesym 2300 Characterization of inequality in terms of reversed equality (see bicom 138). (Contributed by BJ, 7-Jul-2018.)
 |-  ( A  =/=  B  <->  -.  B  =  A )
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