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Theorem necon2d 2667
Description: Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
Hypothesis
Ref Expression
necon2d.1  |-  ( ph  ->  ( A  =  B  ->  C  =/=  D
) )
Assertion
Ref Expression
necon2d  |-  ( ph  ->  ( C  =  D  ->  A  =/=  B
) )

Proof of Theorem necon2d
StepHypRef Expression
1 necon2d.1 . . 3  |-  ( ph  ->  ( A  =  B  ->  C  =/=  D
) )
2 df-ne 2638 . . 3  |-  ( C  =/=  D  <->  -.  C  =  D )
31, 2syl6ib 226 . 2  |-  ( ph  ->  ( A  =  B  ->  -.  C  =  D ) )
43necon2ad 2654 1  |-  ( ph  ->  ( C  =  D  ->  A  =/=  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1381    =/= wne 2636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-ne 2638
This theorem is referenced by:  map0g  7456  cantnf  8110  cantnfOLD  8132  hashprg  12434  bcthlem5  21633  deg1ldgn  22359  cxpeq0  22924  islshpat  34444  cdleme18b  35719  cdlemh  36245
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