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Theorem necon4abiddc 2328
Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)
Hypothesis
Ref Expression
necon4abiddc.1  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
ps  ->  ( A  =/= 
B  <->  -.  ps )
) ) )
Assertion
Ref Expression
necon4abiddc  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
ps  ->  ( A  =  B  <->  ps ) ) ) )

Proof of Theorem necon4abiddc
StepHypRef Expression
1 necon4abiddc.1 . . 3  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
ps  ->  ( A  =/= 
B  <->  -.  ps )
) ) )
2 df-ne 2256 . . . 4  |-  ( A  =/=  B  <->  -.  A  =  B )
32bibi1i 226 . . 3  |-  ( ( A  =/=  B  <->  -.  ps )  <->  ( -.  A  =  B  <->  -.  ps ) )
41, 3syl8ib 164 . 2  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
ps  ->  ( -.  A  =  B  <->  -.  ps )
) ) )
54con4biddc 792 1  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
ps  ->  ( A  =  B  <->  ps ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103  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:  necon4bbiddc  2329  necon4biddc  2330
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