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Theorem necon3d 2350
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 2348 . 2  |-  ( ph  ->  ( C  =/=  D  ->  -.  A  =  B ) )
3 df-ne 2307 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
42, 3syl6ibr 161 1  |-  ( ph  ->  ( C  =/=  D  ->  A  =/=  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1331    =/= wne 2306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604
This theorem depends on definitions:  df-bi 116  df-ne 2307
This theorem is referenced by:  necon3i  2354  pm13.18  2387  ssn0  3400  suppssfv  5971  suppssov1  5972  nnmord  6406  findcard2  6776  findcard2s  6777  addn0nid  8129  nn0n0n1ge2  9114  xnegdi  9644  efne0  11373  divgcdcoprmex  11772
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