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Theorem necon1bbiidc 2388
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon1bbiidc.1  |-  (DECID  A  =  B  ->  ( A  =/=  B  <->  ph ) )
Assertion
Ref Expression
necon1bbiidc  |-  (DECID  A  =  B  ->  ( -.  ph  <->  A  =  B ) )

Proof of Theorem necon1bbiidc
StepHypRef Expression
1 df-ne 2328 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon1bbiidc.1 . . 3  |-  (DECID  A  =  B  ->  ( A  =/=  B  <->  ph ) )
31, 2bitr3id 193 . 2  |-  (DECID  A  =  B  ->  ( -.  A  =  B  <->  ph ) )
43con1biidc 863 1  |-  (DECID  A  =  B  ->  ( -.  ph  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104  DECID wdc 820    = wceq 1335    =/= wne 2327
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 604  ax-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-dc 821  df-ne 2328
This theorem is referenced by:  necon2bbiidc  2392
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