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Theorem necon1ddc 2412
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 2335 . 2  |-  ( C  =/=  D  <->  -.  C  =  D )
2 necon1ddc.1 . . 3  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =/=  B  ->  C  =  D ) ) )
32necon1bddc 2411 . 2  |-  ( ph  ->  (DECID  A  =  B  -> 
( -.  C  =  D  ->  A  =  B ) ) )
41, 3syl7bi 164 1  |-  ( ph  ->  (DECID  A  =  B  -> 
( C  =/=  D  ->  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  DECID wdc 824    = wceq 1342    =/= wne 2334
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 604  ax-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-ne 2335
This theorem is referenced by:  xblss2ps  12971  xblss2  12972
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