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Theorem necon4bbiddc 2380
Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Hypothesis
Ref Expression
necon4bbiddc.1  |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( -.  ps  <->  A  =/=  B ) ) ) )
Assertion
Ref Expression
necon4bbiddc  |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( ps  <->  A  =  B ) ) ) )

Proof of Theorem necon4bbiddc
StepHypRef Expression
1 necon4bbiddc.1 . . . . . 6  |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( -.  ps  <->  A  =/=  B ) ) ) )
2 bicom 139 . . . . . 6  |-  ( ( -.  ps  <->  A  =/=  B )  <->  ( A  =/= 
B  <->  -.  ps )
)
31, 2syl8ib 165 . . . . 5  |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( A  =/= 
B  <->  -.  ps )
) ) )
43com23 78 . . . 4  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
ps  ->  ( A  =/= 
B  <->  -.  ps )
) ) )
54necon4abiddc 2379 . . 3  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
ps  ->  ( A  =  B  <->  ps ) ) ) )
65com23 78 . 2  |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( A  =  B  <->  ps ) ) ) )
7 bicom 139 . 2  |-  ( ( A  =  B  <->  ps )  <->  ( ps  <->  A  =  B
) )
86, 7syl8ib 165 1  |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( ps  <->  A  =  B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104  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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