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Theorem List for Metamath Proof Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcbvab 2401 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 2402* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  { x  |  ph
 }  =  { y  |  ps }
 
Theoremclelab 2403* 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 2404* 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 2405* 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 2406 Extend wff definition to include the not-free predicate for classes.
 wff  F/_ x A
 
Theoremnfcjust 2407* Justification theorem for df-nfc 2408. (Contributed by Mario Carneiro, 13-Oct-2016.)
 |-  ( A. y F/ x  y  e.  A  <->  A. z F/ x  z  e.  A )
 
Definitiondf-nfc 2408* 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 "for all" or something that expands to one (such as "there exists"). It is defined in terms of the not-free predicate df-nf 1532 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 2409* 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 2410* 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 2411* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( F/_ x A  ->  F/ x  y  e.  A )
 
Theoremnfcrii 2412* 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 2413* Consequence of the not-free predicate. (Note that unlike nfcr 2411, 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 2414* 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 2415 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  A  =  B   =>    |-  ( F/_ x A 
 <-> 
 F/_ x B )
 
Theoremnfcxfr 2416 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 2417 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 2418 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 2419* If  x is disjoint from  A, then  x is not free in  A. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A
 
Theoremnfcvd 2420* 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 2421 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x { x  |  ph
 }
 
Theoremnfnfc1 2422  x is bound in  F/_ x A. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x F/_ x A
 
Theoremnfab 2423 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |-  F/_ x { y  | 
 ph }
 
Theoremnfaba1 2424 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x { y  | 
 A. x ph }
 
Theoremnfnfc 2425 Hypothesis builder for  F/_ y A. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   =>    |- 
 F/ x F/_ y A
 
Theoremnfeq 2426 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  A  =  B
 
Theoremnfel 2427 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  A  e.  B
 
Theoremnfeq1 2428* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x  A  =  B
 
Theoremnfel1 2429* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x  A  e.  B
 
Theoremnfeq2 2430* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x B   =>    |- 
 F/ x  A  =  B
 
Theoremnfel2 2431* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x B   =>    |- 
 F/ x  A  e.  B
 
Theoremnfcrd 2432* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/ x  y  e.  A )
 
Theoremnfeqd 2433 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 2434 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 2435 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 2436 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 2437 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 2438 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 2439 Deduction form of dvelimc 2440. (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 2440 Version of dvelim 1956 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 2441 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 2442 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 2443 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 2444 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 2445* 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 2446 Extend wff notation to include inequality.
 wff  A  =/=  B
 
Syntaxwnel 2447 Extend wff notation to include negated membership.
 wff  A  e/  B
 
Definitiondf-ne 2448 Define inequality. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =/=  B  <->  -.  A  =  B )
 
Definitiondf-nel 2449 Define negated membership. (Contributed by NM, 7-Aug-1994.)
 |-  ( A  e/  B  <->  -.  A  e.  B )
 
Theoremnne 2450 Negation of inequality. (Contributed by NM, 9-Jun-2006.)
 |-  ( -.  A  =/=  B  <->  A  =  B )
 
Theoremneirr 2451 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |- 
 -.  A  =/=  A
 
Theoremexmidne 2452 Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.)
 |-  ( A  =  B  \/  A  =/=  B )
 
Theoremnonconne 2453 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
 |- 
 -.  ( A  =  B  /\  A  =/=  B )
 
Theoremneeq1 2454 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
 |-  ( A  =  B  ->  ( A  =/=  C  <->  B  =/=  C ) )
 
Theoremneeq2 2455 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
 |-  ( A  =  B  ->  ( C  =/=  A  <->  C  =/=  B ) )
 
Theoremneeq1i 2456 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
 |-  A  =  B   =>    |-  ( A  =/=  C  <->  B  =/=  C )
 
Theoremneeq2i 2457 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
 |-  A  =  B   =>    |-  ( C  =/=  A  <->  C  =/=  B )
 
Theoremneeq12i 2458 Inference for inequality. (Contributed by NM, 24-Jul-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  =/=  C  <->  B  =/=  D )
 
Theoremneeq1d 2459 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  C ) )
 
Theoremneeq2d 2460 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  =/=  A  <->  C  =/=  B ) )
 
Theoremneeq12d 2461 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  D ) )
 
Theoremneneqd 2462 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  -.  A  =  B )
 
Theoremeqnetri 2463 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =  B   &    |-  B  =/=  C   =>    |-  A  =/=  C
 
Theoremeqnetrd 2464 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremeqnetrri 2465 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =  B   &    |-  A  =/=  C   =>    |-  B  =/=  C
 
Theoremeqnetrrd 2466 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  B  =/=  C )
 
Theoremneeqtri 2467 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =/=  B   &    |-  B  =  C   =>    |-  A  =/=  C
 
Theoremneeqtrd 2468 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremneeqtrri 2469 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =/=  B   &    |-  C  =  B   =>    |-  A  =/=  C
 
Theoremneeqtrrd 2470 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremsyl5eqner 2471 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
 |-  B  =  A   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theorem3netr3d 2472 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 2473 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 2474 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 2475 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 2476 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
 |-  ( A  =  B  <->  ph )   =>    |-  ( A  =/=  B  <->  -.  ph )
 
Theoremnecon3bbii 2477 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
 |-  ( ph  <->  A  =  B )   =>    |-  ( -.  ph  <->  A  =/=  B )
 
Theoremnecon3bii 2478 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
 |-  ( A  =  B  <->  C  =  D )   =>    |-  ( A  =/=  B  <->  C  =/=  D )
 
Theoremnecon3abid 2479 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
 |-  ( ph  ->  ( A  =  B  <->  ps ) )   =>    |-  ( ph  ->  ( A  =/=  B  <->  -.  ps ) )
 
Theoremnecon3bbid 2480 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
 |-  ( ph  ->  ( ps 
 <->  A  =  B ) )   =>    |-  ( ph  ->  ( -.  ps  <->  A  =/=  B ) )
 
Theoremnecon3bid 2481 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 2482 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 2483 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 2484 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
 |-  ( ph  ->  ( A  =  B  ->  C  =  D ) )   =>    |-  ( ph  ->  ( C  =/=  D  ->  A  =/=  B ) )
 
Theoremnecon3i 2485 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)
 |-  ( A  =  B  ->  C  =  D )   =>    |-  ( C  =/=  D  ->  A  =/=  B )
 
Theoremnecon3ai 2486 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 2487 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 2488 Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.)
 |-  ( -.  ph  ->  A  =  B )   =>    |-  ( A  =/=  B 
 ->  ph )
 
Theoremnecon1bi 2489 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 2490 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
 |-  ( A  =/=  B  ->  C  =  D )   =>    |-  ( C  =/=  D  ->  A  =  B )
 
Theoremnecon2ai 2491 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 2492 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
 |-  ( ph  ->  A  =/=  B )   =>    |-  ( A  =  B  ->  -.  ph )
 
Theoremnecon2i 2493 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
 |-  ( A  =  B  ->  C  =/=  D )   =>    |-  ( C  =  D  ->  A  =/=  B )
 
Theoremnecon2ad 2494 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 2495 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
 |-  ( ph  ->  ( ps  ->  A  =/=  B ) )   =>    |-  ( ph  ->  ( A  =  B  ->  -. 
 ps ) )
 
Theoremnecon2d 2496 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
 |-  ( ph  ->  ( A  =  B  ->  C  =/=  D ) )   =>    |-  ( ph  ->  ( C  =  D  ->  A  =/=  B ) )
 
Theoremnecon1abii 2497 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)
 |-  ( -.  ph  <->  A  =  B )   =>    |-  ( A  =/=  B  <->  ph )
 
Theoremnecon1bbii 2498 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)
 |-  ( A  =/=  B  <->  ph )   =>    |-  ( -.  ph  <->  A  =  B )
 
Theoremnecon1abid 2499 Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.)
 |-  ( ph  ->  ( -.  ps  <->  A  =  B ) )   =>    |-  ( ph  ->  ( A  =/=  B  <->  ps ) )
 
Theoremnecon1bbid 2500 Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)
 |-  ( ph  ->  ( A  =/=  B  <->  ps ) )   =>    |-  ( ph  ->  ( -.  ps  <->  A  =  B ) )
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