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Theorem necon2d 2496
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 2448 . . 3  |-  ( C  =/=  D  <->  -.  C  =  D )
31, 2syl6ib 217 . 2  |-  ( ph  ->  ( A  =  B  ->  -.  C  =  D ) )
43necon2ad 2494 1  |-  ( ph  ->  ( C  =  D  ->  A  =/=  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    =/= wne 2446
This theorem is referenced by:  map0g  6807  cantnf  7395  hashprg  11368  bcthlem5  18750  deg1ldgn  19479  cxpeq0  20025  islshpat  29207  cdleme18b  30481  cdlemh  31006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-ne 2448
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