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Theorem List for Metamath Proof Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdrnfc1 2401 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 2402 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 ) )
 
Theoremnfabd2 2403 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ( ph  /\ 
 -.  A. x  x  =  y )  ->  F/ x ps )   =>    |-  ( ph  ->  F/_ x { y  |  ps } )
 
Theoremnfabd 2404 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 2405 Deduction form of dvelimc 2406. (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 2406 Version of dvelim 2092 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 2407 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 2408 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 2409 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (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 2410 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 2411* 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
 
Syntaxwne 2412 Extend wff notation to include inequality.
 wff  A  =/=  B
 
Syntaxwnel 2413 Extend wff notation to include negated membership.
 wff  A  e/  B
 
Definitiondf-ne 2414 Define inequality. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =/=  B  <->  -.  A  =  B )
 
Definitiondf-nel 2415 Define negated membership. (Contributed by NM, 7-Aug-1994.)
 |-  ( A  e/  B  <->  -.  A  e.  B )
 
Theoremnne 2416 Negation of inequality. (Contributed by NM, 9-Jun-2006.)
 |-  ( -.  A  =/=  B  <->  A  =  B )
 
Theoremneirr 2417 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |- 
 -.  A  =/=  A
 
Theoremexmidne 2418 Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.)
 |-  ( A  =  B  \/  A  =/=  B )
 
Theoremnonconne 2419 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
 |- 
 -.  ( A  =  B  /\  A  =/=  B )
 
Theoremneeq1 2420 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
 |-  ( A  =  B  ->  ( A  =/=  C  <->  B  =/=  C ) )
 
Theoremneeq2 2421 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
 |-  ( A  =  B  ->  ( C  =/=  A  <->  C  =/=  B ) )
 
Theoremneeq1i 2422 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
 |-  A  =  B   =>    |-  ( A  =/=  C  <->  B  =/=  C )
 
Theoremneeq2i 2423 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
 |-  A  =  B   =>    |-  ( C  =/=  A  <->  C  =/=  B )
 
Theoremneeq12i 2424 Inference for inequality. (Contributed by NM, 24-Jul-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  =/=  C  <->  B  =/=  D )
 
Theoremneeq1d 2425 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  C ) )
 
Theoremneeq2d 2426 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  =/=  A  <->  C  =/=  B ) )
 
Theoremneeq12d 2427 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  D ) )
 
Theoremneneqd 2428 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  -.  A  =  B )
 
Theoremeqnetri 2429 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =  B   &    |-  B  =/=  C   =>    |-  A  =/=  C
 
Theoremeqnetrd 2430 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremeqnetrri 2431 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =  B   &    |-  A  =/=  C   =>    |-  B  =/=  C
 
Theoremeqnetrrd 2432 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  B  =/=  C )
 
Theoremneeqtri 2433 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =/=  B   &    |-  B  =  C   =>    |-  A  =/=  C
 
Theoremneeqtrd 2434 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremneeqtrri 2435 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =/=  B   &    |-  C  =  B   =>    |-  A  =/=  C
 
Theoremneeqtrrd 2436 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremsyl5eqner 2437 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
 |-  B  =  A   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theorem3netr3d 2438 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 2439 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 2440 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 2441 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 2442 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
 |-  ( A  =  B  <->  ph )   =>    |-  ( A  =/=  B  <->  -.  ph )
 
Theoremnecon3bbii 2443 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
 |-  ( ph  <->  A  =  B )   =>    |-  ( -.  ph  <->  A  =/=  B )
 
Theoremnecon3bii 2444 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
 |-  ( A  =  B  <->  C  =  D )   =>    |-  ( A  =/=  B  <->  C  =/=  D )
 
Theoremnecon3abid 2445 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
 |-  ( ph  ->  ( A  =  B  <->  ps ) )   =>    |-  ( ph  ->  ( A  =/=  B  <->  -.  ps ) )
 
Theoremnecon3bbid 2446 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
 |-  ( ph  ->  ( ps 
 <->  A  =  B ) )   =>    |-  ( ph  ->  ( -.  ps  <->  A  =/=  B ) )
 
Theoremnecon3bid 2447 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 2448 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  ( ps  ->  A  =  B ) )   =>    |-  ( ph  ->  ( A  =/=  B  ->  -.  ps ) )
 
Theoremnecon3bd 2449 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  ( A  =  B  ->  ps ) )   =>    |-  ( ph  ->  ( -.  ps  ->  A  =/=  B ) )
 
Theoremnecon3d 2450 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
 |-  ( ph  ->  ( A  =  B  ->  C  =  D ) )   =>    |-  ( ph  ->  ( C  =/=  D  ->  A  =/=  B ) )
 
Theoremnecon3i 2451 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)
 |-  ( A  =  B  ->  C  =  D )   =>    |-  ( C  =/=  D  ->  A  =/=  B )
 
Theoremnecon3ai 2452 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( A  =/=  B  ->  -.  ph )
 
Theoremnecon3bi 2453 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A  =  B  -> 
 ph )   =>    |-  ( -.  ph  ->  A  =/=  B )
 
Theoremnecon1ai 2454 Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.)
 |-  ( -.  ph  ->  A  =  B )   =>    |-  ( A  =/=  B 
 ->  ph )
 
Theoremnecon1bi 2455 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A  =/=  B  -> 
 ph )   =>    |-  ( -.  ph  ->  A  =  B )
 
Theoremnecon1i 2456 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
 |-  ( A  =/=  B  ->  C  =  D )   =>    |-  ( C  =/=  D  ->  A  =  B )
 
Theoremnecon2ai 2457 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A  =  B  ->  -.  ph )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremnecon2bi 2458 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
 |-  ( ph  ->  A  =/=  B )   =>    |-  ( A  =  B  ->  -.  ph )
 
Theoremnecon2i 2459 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
 |-  ( A  =  B  ->  C  =/=  D )   =>    |-  ( C  =  D  ->  A  =/=  B )
 
Theoremnecon2ad 2460 Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  ( A  =  B  ->  -. 
 ps ) )   =>    |-  ( ph  ->  ( ps  ->  A  =/=  B ) )
 
Theoremnecon2bd 2461 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
 |-  ( ph  ->  ( ps  ->  A  =/=  B ) )   =>    |-  ( ph  ->  ( A  =  B  ->  -. 
 ps ) )
 
Theoremnecon2d 2462 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
 |-  ( ph  ->  ( A  =  B  ->  C  =/=  D ) )   =>    |-  ( ph  ->  ( C  =  D  ->  A  =/=  B ) )
 
Theoremnecon1abii 2463 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)
 |-  ( -.  ph  <->  A  =  B )   =>    |-  ( A  =/=  B  <->  ph )
 
Theoremnecon1bbii 2464 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)
 |-  ( A  =/=  B  <->  ph )   =>    |-  ( -.  ph  <->  A  =  B )
 
Theoremnecon1abid 2465 Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.)
 |-  ( ph  ->  ( -.  ps  <->  A  =  B ) )   =>    |-  ( ph  ->  ( A  =/=  B  <->  ps ) )
 
Theoremnecon1bbid 2466 Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)
 |-  ( ph  ->  ( A  =/=  B  <->  ps ) )   =>    |-  ( ph  ->  ( -.  ps  <->  A  =  B ) )
 
Theoremnecon2abii 2467 Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)
 |-  ( A  =  B  <->  -.  ph )   =>    |-  ( ph  <->  A  =/=  B )
 
Theoremnecon2bbii 2468 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
 |-  ( ph  <->  A  =/=  B )   =>    |-  ( A  =  B  <->  -.  ph )
 
Theoremnecon2abid 2469 Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.)
 |-  ( ph  ->  ( A  =  B  <->  -.  ps ) )   =>    |-  ( ph  ->  ( ps  <->  A  =/=  B ) )
 
Theoremnecon2bbid 2470 Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.)
 |-  ( ph  ->  ( ps 
 <->  A  =/=  B ) )   =>    |-  ( ph  ->  ( A  =  B  <->  -.  ps ) )
 
Theoremnecon4ai 2471 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A  =/=  B  ->  -.  ph )   =>    |-  ( ph  ->  A  =  B )
 
Theoremnecon4i 2472 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A  =/=  B  ->  C  =/=  D )   =>    |-  ( C  =  D  ->  A  =  B )
 
Theoremnecon4ad 2473 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  ( A  =/=  B  ->  -.  ps ) )   =>    |-  ( ph  ->  ( ps  ->  A  =  B ) )
 
Theoremnecon4bd 2474 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  ( -.  ps  ->  A  =/=  B ) )   =>    |-  ( ph  ->  ( A  =  B  ->  ps ) )
 
Theoremnecon4d 2475 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  ( A  =/=  B  ->  C  =/=  D ) )   =>    |-  ( ph  ->  ( C  =  D  ->  A  =  B ) )
 
Theoremnecon4abid 2476 Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.)
 |-  ( ph  ->  ( A  =/=  B  <->  -.  ps ) )   =>    |-  ( ph  ->  ( A  =  B  <->  ps ) )
 
Theoremnecon4bbid 2477 Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)
 |-  ( ph  ->  ( -.  ps  <->  A  =/=  B ) )   =>    |-  ( ph  ->  ( ps 
 <->  A  =  B ) )
 
Theoremnecon4bid 2478 Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)
 |-  ( ph  ->  ( A  =/=  B  <->  C  =/=  D ) )   =>    |-  ( ph  ->  ( A  =  B  <->  C  =  D ) )
 
Theoremnecon1ad 2479 Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.)
 |-  ( ph  ->  ( -.  ps  ->  A  =  B ) )   =>    |-  ( ph  ->  ( A  =/=  B  ->  ps ) )
 
Theoremnecon1bd 2480 Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  ( A  =/=  B  ->  ps )
 )   =>    |-  ( ph  ->  ( -.  ps  ->  A  =  B ) )
 
Theoremnecon1d 2481 Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  ( A  =/=  B  ->  C  =  D ) )   =>    |-  ( ph  ->  ( C  =/=  D  ->  A  =  B ) )
 
Theoremneneqad 2482 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2428. One-way deduction form of df-ne 2414. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  -.  A  =  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremnebi 2483 Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)
 |-  ( ( A  =  B 
 <->  C  =  D )  <-> 
 ( A  =/=  B  <->  C  =/=  D ) )
 
Theorempm13.18 2484 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( ( A  =  B  /\  A  =/=  C )  ->  B  =/=  C )
 
Theorempm13.181 2485 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( ( A  =  B  /\  B  =/=  C )  ->  A  =/=  C )
 
Theorempm2.21ddne 2486 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ps )
 
Theorempm2.61ne 2487 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 ( ph  /\  A  =/=  B )  ->  ps )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  ps )
 
Theorempm2.61ine 2488 Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A  =  B  -> 
 ph )   &    |-  ( A  =/=  B 
 ->  ph )   =>    |-  ph
 
Theorempm2.61dne 2489 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  ( A  =  B  ->  ps ) )   &    |-  ( ph  ->  ( A  =/=  B  ->  ps ) )   =>    |-  ( ph  ->  ps )
 
Theorempm2.61dane 2490 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)
 |-  ( ( ph  /\  A  =  B )  ->  ps )   &    |-  (
 ( ph  /\  A  =/=  B )  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theorempm2.61da2ne 2491 Deduction eliminating two inequalities in an antecedent. (Contributed by NM, 29-May-2013.)
 |-  ( ( ph  /\  A  =  B )  ->  ps )   &    |-  (
 ( ph  /\  C  =  D )  ->  ps )   &    |-  (
 ( ph  /\  ( A  =/=  B  /\  C  =/=  D ) )  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theorempm2.61da3ne 2492 Deduction eliminating three inequalities in an antecedent. (Contributed by NM, 15-Jun-2013.)
 |-  ( ( ph  /\  A  =  B )  ->  ps )   &    |-  (
 ( ph  /\  C  =  D )  ->  ps )   &    |-  (
 ( ph  /\  E  =  F )  ->  ps )   &    |-  (
 ( ph  /\  ( A  =/=  B  /\  C  =/=  D  /\  E  =/=  F ) )  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremnecom 2493 Commutation of inequality. (Contributed by NM, 14-May-1999.)
 |-  ( A  =/=  B  <->  B  =/=  A )
 
Theoremnecomi 2494 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
 |-  A  =/=  B   =>    |-  B  =/=  A
 
Theoremnecomd 2495 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
 |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  B  =/=  A )
 
Theoremneor 2496 Logical OR with an equality. (Contributed by NM, 29-Apr-2007.)
 |-  ( ( A  =  B  \/  ps )  <->  ( A  =/=  B 
 ->  ps ) )
 
Theoremneanior 2497 A DeMorgan's law for inequality. (Contributed by NM, 18-May-2007.)
 |-  ( ( A  =/=  B 
 /\  C  =/=  D ) 
 <->  -.  ( A  =  B  \/  C  =  D ) )
 
Theoremne3anior 2498 A DeMorgan's law for inequality. (Contributed by NM, 30-Sep-2013.)
 |-  ( ( A  =/=  B 
 /\  C  =/=  D  /\  E  =/=  F )  <->  -.  ( A  =  B  \/  C  =  D  \/  E  =  F )
 )
 
Theoremneorian 2499 A DeMorgan's law for inequality. (Contributed by NM, 18-May-2007.)
 |-  ( ( A  =/=  B  \/  C  =/=  D ) 
 <->  -.  ( A  =  B  /\  C  =  D ) )
 
Theoremnemtbir 2500 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
 |-  A  =/=  B   &    |-  ( ph 
 <->  A  =  B )   =>    |-  -.  ph
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