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Theorem necon2abiddc 2374
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2abiddc.1  |-  ( ph  ->  (DECID  ps  ->  ( A  =  B  <->  -.  ps )
) )
Assertion
Ref Expression
necon2abiddc  |-  ( ph  ->  (DECID  ps  ->  ( ps  <->  A  =/=  B ) ) )

Proof of Theorem necon2abiddc
StepHypRef Expression
1 necon2abiddc.1 . . . 4  |-  ( ph  ->  (DECID  ps  ->  ( A  =  B  <->  -.  ps )
) )
2 bicom 139 . . . 4  |-  ( ( A  =  B  <->  -.  ps )  <->  ( -.  ps  <->  A  =  B ) )
31, 2syl6ib 160 . . 3  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps 
<->  A  =  B ) ) )
43necon1abiddc 2370 . 2  |-  ( ph  ->  (DECID  ps  ->  ( A  =/=  B  <->  ps ) ) )
5 bicom 139 . 2  |-  ( ( A  =/=  B  <->  ps )  <->  ( ps  <->  A  =/=  B
) )
64, 5syl6ib 160 1  |-  ( ph  ->  (DECID  ps  ->  ( ps  <->  A  =/=  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104  DECID wdc 819    = wceq 1331    =/= wne 2308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-ne 2309
This theorem is referenced by: (None)
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