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Theorem necon2bbiddc 2414
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2bbiddc.1  |-  ( ph  ->  (DECID  A  =  B  -> 
( ps  <->  A  =/=  B ) ) )
Assertion
Ref Expression
necon2bbiddc  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =  B  <->  -.  ps ) ) )

Proof of Theorem necon2bbiddc
StepHypRef Expression
1 necon2bbiddc.1 . . . 4  |-  ( ph  ->  (DECID  A  =  B  -> 
( ps  <->  A  =/=  B ) ) )
2 bicom 140 . . . 4  |-  ( ( ps  <->  A  =/=  B
)  <->  ( A  =/= 
B  <->  ps ) )
31, 2syl6ib 161 . . 3  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =/=  B  <->  ps ) ) )
43necon1bbiddc 2410 . 2  |-  ( ph  ->  (DECID  A  =  B  -> 
( -.  ps  <->  A  =  B ) ) )
5 bicom 140 . 2  |-  ( ( -.  ps  <->  A  =  B )  <->  ( A  =  B  <->  -.  ps )
)
64, 5syl6ib 161 1  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =  B  <->  -.  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 105  DECID wdc 834    = wceq 1353    =/= wne 2347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-ne 2348
This theorem is referenced by: (None)
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