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Theorem pm2.85dc 845
Description: Reverse distribution of disjunction over implication, given decidability. Based on theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by Jim Kingdon, 1-Apr-2018.)
Assertion
Ref Expression
pm2.85dc  |-  (DECID  ph  ->  ( ( ( ph  \/  ps )  ->  ( ph  \/  ch ) )  -> 
( ph  \/  ( ps  ->  ch ) ) ) )

Proof of Theorem pm2.85dc
StepHypRef Expression
1 df-dc 777 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 orc 666 . . . 4  |-  ( ph  ->  ( ph  \/  ( ps  ->  ch ) ) )
32a1d 22 . . 3  |-  ( ph  ->  ( ( ( ph  \/  ps )  ->  ( ph  \/  ch ) )  ->  ( ph  \/  ( ps  ->  ch )
) ) )
4 olc 665 . . . . . 6  |-  ( ps 
->  ( ph  \/  ps ) )
54imim1i 59 . . . . 5  |-  ( ( ( ph  \/  ps )  ->  ( ph  \/  ch ) )  ->  ( ps  ->  ( ph  \/  ch ) ) )
6 orel1 677 . . . . 5  |-  ( -. 
ph  ->  ( ( ph  \/  ch )  ->  ch ) )
75, 6syl9r 72 . . . 4  |-  ( -. 
ph  ->  ( ( (
ph  \/  ps )  ->  ( ph  \/  ch ) )  ->  ( ps  ->  ch ) ) )
8 olc 665 . . . 4  |-  ( ( ps  ->  ch )  ->  ( ph  \/  ( ps  ->  ch ) ) )
97, 8syl6 33 . . 3  |-  ( -. 
ph  ->  ( ( (
ph  \/  ps )  ->  ( ph  \/  ch ) )  ->  ( ph  \/  ( ps  ->  ch ) ) ) )
103, 9jaoi 669 . 2  |-  ( (
ph  \/  -.  ph )  ->  ( ( ( ph  \/  ps )  ->  ( ph  \/  ch ) )  ->  ( ph  \/  ( ps  ->  ch )
) ) )
111, 10sylbi 119 1  |-  (DECID  ph  ->  ( ( ( ph  \/  ps )  ->  ( ph  \/  ch ) )  -> 
( ph  \/  ( ps  ->  ch ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 662  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-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by:  orimdidc  846
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