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Theorem necon1addc 2331
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Hypothesis
Ref Expression
necon1addc.1  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  A  =  B ) ) )
Assertion
Ref Expression
necon1addc  |-  ( ph  ->  (DECID  ps  ->  ( A  =/=  B  ->  ps )
) )

Proof of Theorem necon1addc
StepHypRef Expression
1 df-ne 2256 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon1addc.1 . . 3  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  A  =  B ) ) )
3 con1dc 791 . . 3  |-  (DECID  ps  ->  ( ( -.  ps  ->  A  =  B )  -> 
( -.  A  =  B  ->  ps )
) )
42, 3sylcom 28 . 2  |-  ( ph  ->  (DECID  ps  ->  ( -.  A  =  B  ->  ps ) ) )
51, 4syl7bi 163 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-in1 579  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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