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Theorem pm2.54dc 828
Description: Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 676, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.)
Assertion
Ref Expression
pm2.54dc  |-  (DECID  ph  ->  ( ( -.  ph  ->  ps )  ->  ( ph  \/  ps ) ) )

Proof of Theorem pm2.54dc
StepHypRef Expression
1 dcn 784 . 2  |-  (DECID  ph  -> DECID  -.  ph )
2 notnotrdc 789 . . . . 5  |-  (DECID  ph  ->  ( -.  -.  ph  ->  ph ) )
3 orc 668 . . . . 5  |-  ( ph  ->  ( ph  \/  ps ) )
42, 3syl6 33 . . . 4  |-  (DECID  ph  ->  ( -.  -.  ph  ->  (
ph  \/  ps )
) )
54a1d 22 . . 3  |-  (DECID  ph  ->  (DECID  -. 
ph  ->  ( -.  -.  ph 
->  ( ph  \/  ps ) ) ) )
6 olc 667 . . . 4  |-  ( ps 
->  ( ph  \/  ps ) )
76a1i 9 . . 3  |-  (DECID  ph  ->  ( ps  ->  ( ph  \/  ps ) ) )
85, 7jaddc 799 . 2  |-  (DECID  ph  ->  (DECID  -. 
ph  ->  ( ( -. 
ph  ->  ps )  -> 
( ph  \/  ps ) ) ) )
91, 8mpd 13 1  |-  (DECID  ph  ->  ( ( -.  ph  ->  ps )  ->  ( ph  \/  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 664  DECID wdc 780
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 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by:  dfordc  829  pm2.68dc  831  pm4.79dc  847  pm5.11dc  853
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