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Theorem necon4addc 2353
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
Hypothesis
Ref Expression
necon4addc.1  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =/=  B  ->  -.  ps ) ) )
Assertion
Ref Expression
necon4addc  |-  ( ph  ->  (DECID  A  =  B  -> 
( ps  ->  A  =  B ) ) )

Proof of Theorem necon4addc
StepHypRef Expression
1 necon4addc.1 . 2  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =/=  B  ->  -.  ps ) ) )
2 df-ne 2284 . . . 4  |-  ( A  =/=  B  <->  -.  A  =  B )
32imbi1i 237 . . 3  |-  ( ( A  =/=  B  ->  -.  ps )  <->  ( -.  A  =  B  ->  -. 
ps ) )
4 condc 821 . . 3  |-  (DECID  A  =  B  ->  ( ( -.  A  =  B  ->  -.  ps )  -> 
( ps  ->  A  =  B ) ) )
53, 4syl5bi 151 . 2  |-  (DECID  A  =  B  ->  ( ( A  =/=  B  ->  -.  ps )  ->  ( ps 
->  A  =  B
) ) )
61, 5sylcom 28 1  |-  ( ph  ->  (DECID  A  =  B  -> 
( ps  ->  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  DECID wdc 802    = wceq 1314    =/= wne 2283
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 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-ne 2284
This theorem is referenced by: (None)
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