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Theorem necon3d 2264
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 2262 . 2  |-  ( ph  ->  ( C  =/=  D  ->  -.  A  =  B ) )
3 df-ne 2221 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
42, 3syl6ibr 155 1  |-  ( ph  ->  ( C  =/=  D  ->  A  =/=  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1259    =/= wne 2220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555
This theorem depends on definitions:  df-bi 114  df-ne 2221
This theorem is referenced by:  necon3i  2268  pm13.18  2301  ssn0  3287  suppssfv  5736  suppssov1  5737  nnmord  6121  findcard2  6377  findcard2s  6378  addn0nid  7444  nn0n0n1ge2  8369
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