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Theorem necon1abiidc 2315
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon1abiidc.1  |-  (DECID  ph  ->  ( -.  ph  <->  A  =  B
) )
Assertion
Ref Expression
necon1abiidc  |-  (DECID  ph  ->  ( A  =/=  B  <->  ph ) )

Proof of Theorem necon1abiidc
StepHypRef Expression
1 df-ne 2256 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon1abiidc.1 . . 3  |-  (DECID  ph  ->  ( -.  ph  <->  A  =  B
) )
32con1biidc 809 . 2  |-  (DECID  ph  ->  ( -.  A  =  B  <->  ph ) )
41, 3syl5bb 190 1  |-  (DECID  ph  ->  ( A  =/=  B  <->  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103  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:  necon2abiidc  2319
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