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Theorem pm5.62dc 863
Description: Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
Assertion
Ref Expression
pm5.62dc  |-  (DECID  ps  ->  ( ( ( ph  /\  ps )  \/  -.  ps )  <->  ( ph  \/  -.  ps ) ) )

Proof of Theorem pm5.62dc
StepHypRef Expression
1 df-dc 754 . 2  |-  (DECID  ps  <->  ( ps  \/  -.  ps ) )
2 ordir 741 . . . 4  |-  ( ( ( ph  /\  ps )  \/  -.  ps )  <->  ( ( ph  \/  -.  ps )  /\  ( ps  \/  -.  ps )
) )
32simplbi 263 . . 3  |-  ( ( ( ph  /\  ps )  \/  -.  ps )  ->  ( ph  \/  -.  ps ) )
42simplbi2 371 . . . 4  |-  ( (
ph  \/  -.  ps )  ->  ( ( ps  \/  -.  ps )  ->  (
( ph  /\  ps )  \/  -.  ps ) ) )
54com12 30 . . 3  |-  ( ( ps  \/  -.  ps )  ->  ( ( ph  \/  -.  ps )  -> 
( ( ph  /\  ps )  \/  -.  ps ) ) )
63, 5impbid2 135 . 2  |-  ( ( ps  \/  -.  ps )  ->  ( ( (
ph  /\  ps )  \/  -.  ps )  <->  ( ph  \/  -.  ps ) ) )
71, 6sylbi 118 1  |-  (DECID  ps  ->  ( ( ( ph  /\  ps )  \/  -.  ps )  <->  ( ph  \/  -.  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by: (None)
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