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Theorem pm4.83dc 897
Description: Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 800, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)
Assertion
Ref Expression
pm4.83dc  |-  (DECID  ph  ->  ( ( ( ph  ->  ps )  /\  ( -. 
ph  ->  ps ) )  <->  ps ) )

Proof of Theorem pm4.83dc
StepHypRef Expression
1 df-dc 781 . . 3  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 pm3.44 670 . . . 4  |-  ( ( ( ph  ->  ps )  /\  ( -.  ph  ->  ps ) )  -> 
( ( ph  \/  -.  ph )  ->  ps ) )
32com12 30 . . 3  |-  ( (
ph  \/  -.  ph )  ->  ( ( ( ph  ->  ps )  /\  ( -.  ph  ->  ps )
)  ->  ps )
)
41, 3sylbi 119 . 2  |-  (DECID  ph  ->  ( ( ( ph  ->  ps )  /\  ( -. 
ph  ->  ps ) )  ->  ps ) )
5 ax-1 5 . . 3  |-  ( ps 
->  ( ph  ->  ps ) )
6 ax-1 5 . . 3  |-  ( ps 
->  ( -.  ph  ->  ps ) )
75, 6jca 300 . 2  |-  ( ps 
->  ( ( ph  ->  ps )  /\  ( -. 
ph  ->  ps ) ) )
84, 7impbid1 140 1  |-  (DECID  ph  ->  ( ( ( ph  ->  ps )  /\  ( -. 
ph  ->  ps ) )  <->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ 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-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by: (None)
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