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Theorem necon3d 2249
Description: Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
Hypothesis
Ref Expression
necon3d.1  |-  ( ph  ->  ( A  =  B  ->  C  =  D ) )
Assertion
Ref Expression
necon3d  |-  ( ph  ->  ( C  =/=  D  ->  A  =/=  B ) )

Proof of Theorem necon3d
StepHypRef Expression
1 necon3d.1 . . 3  |-  ( ph  ->  ( A  =  B  ->  C  =  D ) )
21necon3ad 2247 . 2  |-  ( ph  ->  ( C  =/=  D  ->  -.  A  =  B ) )
3 df-ne 2206 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
42, 3syl6ibr 151 1  |-  ( ph  ->  ( C  =/=  D  ->  A  =/=  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1243    =/= wne 2204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545
This theorem depends on definitions:  df-bi 110  df-ne 2206
This theorem is referenced by:  necon3i  2253  pm13.18  2286  ssn0  3259  suppssfv  5708  suppssov1  5709  nnmord  6090  findcard2  6346  findcard2s  6347  nn0n0n1ge2  8311
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