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Theorem necon4idc 2375
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon4idc.1  |-  (DECID  A  =  B  ->  ( A  =/=  B  ->  C  =/=  D ) )
Assertion
Ref Expression
necon4idc  |-  (DECID  A  =  B  ->  ( C  =  D  ->  A  =  B ) )

Proof of Theorem necon4idc
StepHypRef Expression
1 necon4idc.1 . . 3  |-  (DECID  A  =  B  ->  ( A  =/=  B  ->  C  =/=  D ) )
2 df-ne 2307 . . 3  |-  ( C  =/=  D  <->  -.  C  =  D )
31, 2syl6ib 160 . 2  |-  (DECID  A  =  B  ->  ( A  =/=  B  ->  -.  C  =  D ) )
43necon4aidc 2374 1  |-  (DECID  A  =  B  ->  ( C  =  D  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  DECID wdc 819    = wceq 1331    =/= wne 2306
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 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-ne 2307
This theorem is referenced by: (None)
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