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Theorem dcbi 878
Description: An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
Assertion
Ref Expression
dcbi  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  <->  ps ) ) )

Proof of Theorem dcbi
StepHypRef Expression
1 dcim 818 . . 3  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  ->  ps )
) )
2 dcim 818 . . . 4  |-  (DECID  ps  ->  (DECID  ph  -> DECID  ( ps  ->  ph ) ) )
32com12 30 . . 3  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ps  ->  ph )
) )
4 dcan 876 . . 3  |-  (DECID  ( ph  ->  ps )  ->  (DECID  ( ps  ->  ph )  -> DECID  ( ( ph  ->  ps )  /\  ( ps 
->  ph ) ) ) )
51, 3, 4syl6c 65 . 2  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ( ph  ->  ps )  /\  ( ps 
->  ph ) ) ) )
6 dfbi2 380 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
76dcbii 781 . 2  |-  (DECID  ( ph  <->  ps )  <-> DECID  ( ( ph  ->  ps )  /\  ( ps 
->  ph ) ) )
85, 7syl6ibr 160 1  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  <->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103  DECID wdc 776
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 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by:  xor3dc  1319  pm5.15dc  1321  bilukdc  1328  xordidc  1331
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