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Theorem necon4addc 2325
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 2256 . . . 4  |-  ( A  =/=  B  <->  -.  A  =  B )
32imbi1i 236 . . 3  |-  ( ( A  =/=  B  ->  -.  ps )  <->  ( -.  A  =  B  ->  -. 
ps ) )
4 condc 787 . . 3  |-  (DECID  A  =  B  ->  ( ( -.  A  =  B  ->  -.  ps )  -> 
( ps  ->  A  =  B ) ) )
53, 4syl5bi 150 . 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 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: (None)
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