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Theorem pm5.63dc 888
Description: Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
Assertion
Ref Expression
pm5.63dc  |-  (DECID  ph  ->  ( ( ph  \/  ps ) 
<->  ( ph  \/  ( -.  ph  /\  ps )
) ) )

Proof of Theorem pm5.63dc
StepHypRef Expression
1 df-dc 777 . . 3  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 ordi 763 . . . 4  |-  ( (
ph  \/  ( -.  ph 
/\  ps ) )  <->  ( ( ph  \/  -.  ph )  /\  ( ph  \/  ps ) ) )
32simplbi2 377 . . 3  |-  ( (
ph  \/  -.  ph )  ->  ( ( ph  \/  ps )  ->  ( ph  \/  ( -.  ph  /\  ps ) ) ) )
41, 3sylbi 119 . 2  |-  (DECID  ph  ->  ( ( ph  \/  ps )  ->  ( ph  \/  ( -.  ph  /\  ps ) ) ) )
52simprbi 269 . 2  |-  ( (
ph  \/  ( -.  ph 
/\  ps ) )  -> 
( ph  \/  ps ) )
64, 5impbid1 140 1  |-  (DECID  ph  ->  ( ( ph  \/  ps ) 
<->  ( ph  \/  ( -.  ph  /\  ps )
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ 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-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
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