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Theorem necon4aidc 2323
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon4aidc.1  |-  (DECID  A  =  B  ->  ( A  =/=  B  ->  -.  ph )
)
Assertion
Ref Expression
necon4aidc  |-  (DECID  A  =  B  ->  ( ph  ->  A  =  B ) )

Proof of Theorem necon4aidc
StepHypRef Expression
1 df-ne 2256 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon4aidc.1 . . 3  |-  (DECID  A  =  B  ->  ( A  =/=  B  ->  -.  ph )
)
31, 2syl5bir 151 . 2  |-  (DECID  A  =  B  ->  ( -.  A  =  B  ->  -. 
ph ) )
4 condc 787 . 2  |-  (DECID  A  =  B  ->  ( ( -.  A  =  B  ->  -.  ph )  -> 
( ph  ->  A  =  B ) ) )
53, 4mpd 13 1  |-  (DECID  A  =  B  ->  ( ph  ->  A  =  B ) )
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-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781  df-ne 2256
This theorem is referenced by:  necon4idc  2324
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