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Theorem List for Metamath Proof Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabeq2 2401* Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that abbi 2406 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable.

Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable  ph (that has a free variable  x) to a theorem with a class variable  A, we substitute  x  e.  A for  ph throughout and simplify, where  A is a new class variable not already in the wff. An example is the conversion of zfauscl 4159 to inex1 4171 (look at the instance of zfauscl 4159 that occurs in the proof of inex1 4171). Conversely, to convert a theorem with a class variable  A to one with 
ph, we substitute  { x  | 
ph } for  A throughout and simplify, where  x and  ph are new set and wff variables not already in the wff. An example is cp 7577, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 7576. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 5-Aug-1993.)

 |-  ( A  =  { x  |  ph }  <->  A. x ( x  e.  A  <->  ph ) )
 
Theoremabeq1 2402* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
 |-  ( { x  |  ph
 }  =  A  <->  A. x ( ph  <->  x  e.  A ) )
 
Theoremabeq2i 2403 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)
 |-  A  =  { x  |  ph }   =>    |-  ( x  e.  A  <->  ph )
 
Theoremabeq1i 2404 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)
 |- 
 { x  |  ph }  =  A   =>    |-  ( ph  <->  x  e.  A )
 
Theoremabeq2d 2405 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
 |-  ( ph  ->  A  =  { x  |  ps } )   =>    |-  ( ph  ->  ( x  e.  A  <->  ps ) )
 
Theoremabbi 2406 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 2407* Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  A  <->  ph )   =>    |-  A  =  { x  |  ph }
 
Theoremabbii 2408 Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   =>    |- 
 { x  |  ph }  =  { x  |  ps }
 
Theoremabbid 2409 Equivalent wff's yield equal class abstractions (deduction rule). (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 2410* Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
 
Theoremabbi2dv 2411* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
 |-  ( ph  ->  ( x  e.  A  <->  ps ) )   =>    |-  ( ph  ->  A  =  { x  |  ps } )
 
Theoremabbi1dv 2412* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
 |-  ( ph  ->  ( ps 
 <->  x  e.  A ) )   =>    |-  ( ph  ->  { x  |  ps }  =  A )
 
Theoremabid2 2413* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
 |- 
 { x  |  x  e.  A }  =  A
 
Theoremcbvab 2414 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 2415* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  { x  |  ph
 }  =  { y  |  ps }
 
Theoremclelab 2416* 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 2417* 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 2418* 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 2419 Extend wff definition to include the not-free predicate for classes.
 wff  F/_ x A
 
Theoremnfcjust 2420* Justification theorem for df-nfc 2421. (Contributed by Mario Carneiro, 13-Oct-2016.)
 |-  ( A. y F/ x  y  e.  A  <->  A. z F/ x  z  e.  A )
 
Definitiondf-nfc 2421* 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 1535 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 2422* 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 2423* 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 2424* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( F/_ x A  ->  F/ x  y  e.  A )
 
Theoremnfcrii 2425* 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 2426* Consequence of the not-free predicate. (Note that unlike nfcr 2424, 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 2427* 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 2428 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  A  =  B   =>    |-  ( F/_ x A 
 <-> 
 F/_ x B )
 
Theoremnfcxfr 2429 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 2430 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 2431 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 2432* If  x is disjoint from  A, then  x is not free in  A. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A
 
Theoremnfcvd 2433* 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 2434 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x { x  |  ph
 }
 
Theoremnfnfc1 2435  x is bound in  F/_ x A. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x F/_ x A
 
Theoremnfab 2436 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |-  F/_ x { y  | 
 ph }
 
Theoremnfaba1 2437 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x { y  | 
 A. x ph }
 
Theoremnfnfc 2438 Hypothesis builder for  F/_ y A. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   =>    |- 
 F/ x F/_ y A
 
Theoremnfeq 2439 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  A  =  B
 
Theoremnfel 2440 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  A  e.  B
 
Theoremnfeq1 2441* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x  A  =  B
 
Theoremnfel1 2442* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x  A  e.  B
 
Theoremnfeq2 2443* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x B   =>    |- 
 F/ x  A  =  B
 
Theoremnfel2 2444* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x B   =>    |- 
 F/ x  A  e.  B
 
Theoremnfcrd 2445* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/ x  y  e.  A )
 
Theoremnfeqd 2446 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 2447 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 2448 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 2449 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 2450 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 2451 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 2452 Deduction form of dvelimc 2453. (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 2453 Version of dvelim 1969 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 2454 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 2455 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 2456 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 2457 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 2458* 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 2459 Extend wff notation to include inequality.
 wff  A  =/=  B
 
Syntaxwnel 2460 Extend wff notation to include negated membership.
 wff  A  e/  B
 
Definitiondf-ne 2461 Define inequality. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =/=  B  <->  -.  A  =  B )
 
Definitiondf-nel 2462 Define negated membership. (Contributed by NM, 7-Aug-1994.)
 |-  ( A  e/  B  <->  -.  A  e.  B )
 
Theoremnne 2463 Negation of inequality. (Contributed by NM, 9-Jun-2006.)
 |-  ( -.  A  =/=  B  <->  A  =  B )
 
Theoremneirr 2464 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |- 
 -.  A  =/=  A
 
Theoremexmidne 2465 Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.)
 |-  ( A  =  B  \/  A  =/=  B )
 
Theoremnonconne 2466 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
 |- 
 -.  ( A  =  B  /\  A  =/=  B )
 
Theoremneeq1 2467 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
 |-  ( A  =  B  ->  ( A  =/=  C  <->  B  =/=  C ) )
 
Theoremneeq2 2468 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
 |-  ( A  =  B  ->  ( C  =/=  A  <->  C  =/=  B ) )
 
Theoremneeq1i 2469 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
 |-  A  =  B   =>    |-  ( A  =/=  C  <->  B  =/=  C )
 
Theoremneeq2i 2470 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
 |-  A  =  B   =>    |-  ( C  =/=  A  <->  C  =/=  B )
 
Theoremneeq12i 2471 Inference for inequality. (Contributed by NM, 24-Jul-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  =/=  C  <->  B  =/=  D )
 
Theoremneeq1d 2472 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  C ) )
 
Theoremneeq2d 2473 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  =/=  A  <->  C  =/=  B ) )
 
Theoremneeq12d 2474 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  D ) )
 
Theoremneneqd 2475 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  -.  A  =  B )
 
Theoremeqnetri 2476 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =  B   &    |-  B  =/=  C   =>    |-  A  =/=  C
 
Theoremeqnetrd 2477 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremeqnetrri 2478 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =  B   &    |-  A  =/=  C   =>    |-  B  =/=  C
 
Theoremeqnetrrd 2479 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  B  =/=  C )
 
Theoremneeqtri 2480 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =/=  B   &    |-  B  =  C   =>    |-  A  =/=  C
 
Theoremneeqtrd 2481 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremneeqtrri 2482 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  A  =/=  B   &    |-  C  =  B   =>    |-  A  =/=  C
 
Theoremneeqtrrd 2483 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremsyl5eqner 2484 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
 |-  B  =  A   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  A  =/=  C )
 
Theorem3netr3d 2485 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 2486 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 2487 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 2488 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 2489 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
 |-  ( A  =  B  <->  ph )   =>    |-  ( A  =/=  B  <->  -.  ph )
 
Theoremnecon3bbii 2490 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
 |-  ( ph  <->  A  =  B )   =>    |-  ( -.  ph  <->  A  =/=  B )
 
Theoremnecon3bii 2491 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
 |-  ( A  =  B  <->  C  =  D )   =>    |-  ( A  =/=  B  <->  C  =/=  D )
 
Theoremnecon3abid 2492 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
 |-  ( ph  ->  ( A  =  B  <->  ps ) )   =>    |-  ( ph  ->  ( A  =/=  B  <->  -.  ps ) )
 
Theoremnecon3bbid 2493 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
 |-  ( ph  ->  ( ps 
 <->  A  =  B ) )   =>    |-  ( ph  ->  ( -.  ps  <->  A  =/=  B ) )
 
Theoremnecon3bid 2494 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 2495 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 2496 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 2497 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
 |-  ( ph  ->  ( A  =  B  ->  C  =  D ) )   =>    |-  ( ph  ->  ( C  =/=  D  ->  A  =/=  B ) )
 
Theoremnecon3i 2498 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)
 |-  ( A  =  B  ->  C  =  D )   =>    |-  ( C  =/=  D  ->  A  =/=  B )
 
Theoremnecon3ai 2499 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 2500 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A  =  B  -> 
 ph )   =>    |-  ( -.  ph  ->  A  =/=  B )
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