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Theorem necon2d 2314
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 2256 . . 3  |-  ( C  =/=  D  <->  -.  C  =  D )
31, 2syl6ib 159 . 2  |-  ( ph  ->  ( A  =  B  ->  -.  C  =  D ) )
43necon2ad 2312 1  |-  ( ph  ->  ( C  =  D  ->  A  =/=  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1289    =/= wne 2255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580
This theorem depends on definitions:  df-bi 115  df-ne 2256
This theorem is referenced by:  map0g  6425  hashprg  10181
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