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Theorem necon3ai 2334
Description: Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon3ai.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
necon3ai  |-  ( A  =/=  B  ->  -.  ph )

Proof of Theorem necon3ai
StepHypRef Expression
1 df-ne 2286 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon3ai.1 . . 3  |-  ( ph  ->  A  =  B )
32con3i 606 . 2  |-  ( -.  A  =  B  ->  -.  ph )
41, 3sylbi 120 1  |-  ( A  =/=  B  ->  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1316    =/= wne 2285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 588  ax-in2 589
This theorem depends on definitions:  df-bi 116  df-ne 2286
This theorem is referenced by:  disjsn2  3556  0nelxp  4537  fvunsng  5582  map0b  6549  difinfsnlem  6952  hashprg  10522  gcd1  11602  gcdzeq  11637  phimullem  11828
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