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Theorem necon3bd 2292
Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon3bd.1  |-  ( ph  ->  ( A  =  B  ->  ps ) )
Assertion
Ref Expression
necon3bd  |-  ( ph  ->  ( -.  ps  ->  A  =/=  B ) )

Proof of Theorem necon3bd
StepHypRef Expression
1 necon3bd.1 . . 3  |-  ( ph  ->  ( A  =  B  ->  ps ) )
21con3d 594 . 2  |-  ( ph  ->  ( -.  ps  ->  -.  A  =  B ) )
3 df-ne 2250 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
42, 3syl6ibr 160 1  |-  ( ph  ->  ( -.  ps  ->  A  =/=  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1285    =/= wne 2249
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 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-ne 2250
This theorem is referenced by:  nelne1  2339  nelne2  2340  nssne1  3066  nssne2  3067  disjne  3318  difsn  3548  nbrne1  3828  nbrne2  3829  ac6sfi  6544  indpi  6804  zneo  8743
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