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Theorem con1bdc 816
Description: Contraposition. Bidirectional version of con1dc 797. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
con1bdc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  ->  ps )  <->  ( -.  ps  ->  ph ) ) ) )

Proof of Theorem con1bdc
StepHypRef Expression
1 con1dc 797 . . . 4  |-  (DECID  ph  ->  ( ( -.  ph  ->  ps )  ->  ( -.  ps  ->  ph ) ) )
21adantr 272 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ph  ->  ps )  -> 
( -.  ps  ->  ph ) ) )
3 con1dc 797 . . . 4  |-  (DECID  ps  ->  ( ( -.  ps  ->  ph )  ->  ( -.  ph 
->  ps ) ) )
43adantl 273 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ps  ->  ph )  ->  ( -.  ph  ->  ps )
) )
52, 4impbid 128 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ph  ->  ps )  <->  ( -.  ps  ->  ph ) ) )
65ex 114 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  ->  ps )  <->  ( -.  ps  ->  ph ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671
This theorem depends on definitions:  df-bi 116  df-dc 787
This theorem is referenced by: (None)
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