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Theorem pm5.11dc 853
Description: A decidable proposition or its negation implies a second proposition. Based on theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 29-Mar-2018.)
Assertion
Ref Expression
pm5.11dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  ->  ps )  \/  ( -. 
ph  ->  ps ) ) ) )

Proof of Theorem pm5.11dc
StepHypRef Expression
1 dcim 822 . 2  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  ->  ps )
) )
2 pm2.5dc 801 . . 3  |-  (DECID  ph  ->  ( -.  ( ph  ->  ps )  ->  ( -.  ph 
->  ps ) ) )
3 pm2.54dc 828 . . 3  |-  (DECID  ( ph  ->  ps )  ->  (
( -.  ( ph  ->  ps )  ->  ( -.  ph  ->  ps )
)  ->  ( ( ph  ->  ps )  \/  ( -.  ph  ->  ps ) ) ) )
42, 3syl5com 29 . 2  |-  (DECID  ph  ->  (DECID  (
ph  ->  ps )  -> 
( ( ph  ->  ps )  \/  ( -. 
ph  ->  ps ) ) ) )
51, 4syld 44 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( 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: (None)
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