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Theorem necon1ddc 2333
Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Hypothesis
Ref Expression
necon1ddc.1  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =/=  B  ->  C  =  D ) ) )
Assertion
Ref Expression
necon1ddc  |-  ( ph  ->  (DECID  A  =  B  -> 
( C  =/=  D  ->  A  =  B ) ) )

Proof of Theorem necon1ddc
StepHypRef Expression
1 df-ne 2256 . 2  |-  ( C  =/=  D  <->  -.  C  =  D )
2 necon1ddc.1 . . 3  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =/=  B  ->  C  =  D ) ) )
32necon1bddc 2332 . 2  |-  ( ph  ->  (DECID  A  =  B  -> 
( -.  C  =  D  ->  A  =  B ) ) )
41, 3syl7bi 163 1  |-  ( ph  ->  (DECID  A  =  B  -> 
( C  =/=  D  ->  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-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: (None)
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