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Theorem necon4bddc 2326
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
Hypothesis
Ref Expression
necon4bddc.1  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  A  =/=  B
) ) )
Assertion
Ref Expression
necon4bddc  |-  ( ph  ->  (DECID  ps  ->  ( A  =  B  ->  ps )
) )

Proof of Theorem necon4bddc
StepHypRef Expression
1 necon4bddc.1 . . 3  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  A  =/=  B
) ) )
2 df-ne 2256 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
31, 2syl8ib 164 . 2  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  -.  A  =  B ) ) )
4 condc 787 . 2  |-  (DECID  ps  ->  ( ( -.  ps  ->  -.  A  =  B )  ->  ( A  =  B  ->  ps )
) )
53, 4sylcom 28 1  |-  ( ph  ->  (DECID  ps  ->  ( A  =  B  ->  ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  DECID wdc 780    = wceq 1289    =/= wne 2255
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-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781  df-ne 2256
This theorem is referenced by: (None)
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