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Theorem pm3.12dc 900
Description: Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
Assertion
Ref Expression
pm3.12dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  \/  -.  ps )  \/  ( ph  /\  ps ) ) ) )

Proof of Theorem pm3.12dc
StepHypRef Expression
1 pm3.11dc 899 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( -. 
ph  \/  -.  ps )  ->  ( ph  /\  ps ) ) ) )
21imp 122 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( -.  ph  \/  -.  ps )  ->  ( ph  /\  ps ) ) )
3 dcn 780 . . . . . 6  |-  (DECID  ph  -> DECID  -.  ph )
4 dcn 780 . . . . . 6  |-  (DECID  ps  -> DECID  -.  ps )
5 dcor 877 . . . . . 6  |-  (DECID  -.  ph  ->  (DECID  -.  ps  -> DECID  ( -.  ph  \/  -.  ps ) ) )
63, 4, 5syl2im 38 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( -.  ph  \/  -.  ps ) ) )
7 dfordc 825 . . . . 5  |-  (DECID  ( -. 
ph  \/  -.  ps )  ->  ( ( ( -. 
ph  \/  -.  ps )  \/  ( ph  /\  ps ) )  <->  ( -.  ( -.  ph  \/  -.  ps )  ->  ( ph  /\ 
ps ) ) ) )
86, 7syl6 33 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ( -. 
ph  \/  -.  ps )  \/  ( ph  /\  ps ) )  <->  ( -.  ( -.  ph  \/  -.  ps )  ->  ( ph  /\ 
ps ) ) ) ) )
98imp 122 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ( -.  ph  \/  -.  ps )  \/  ( ph  /\  ps ) )  <-> 
( -.  ( -. 
ph  \/  -.  ps )  ->  ( ph  /\  ps ) ) ) )
102, 9mpbird 165 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ph  \/  -.  ps )  \/  ( ph  /\  ps ) ) )
1110ex 113 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  \/  -.  ps )  \/  ( 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-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
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