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Theorem pm3.12dc 927
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 926 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( -. 
ph  \/  -.  ps )  ->  ( ph  /\  ps ) ) ) )
21imp 123 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( -.  ph  \/  -.  ps )  ->  ( ph  /\  ps ) ) )
3 dcn 812 . . . . . 6  |-  (DECID  ph  -> DECID  -.  ph )
4 dcn 812 . . . . . 6  |-  (DECID  ps  -> DECID  -.  ps )
5 dcor 904 . . . . . 6  |-  (DECID  -.  ph  ->  (DECID  -.  ps  -> DECID  ( -.  ph  \/  -.  ps ) ) )
63, 4, 5syl2im 38 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( -.  ph  \/  -.  ps ) ) )
7 dfordc 862 . . . . 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 123 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ( -.  ph  \/  -.  ps )  \/  ( ph  /\  ps ) )  <-> 
( -.  ( -. 
ph  \/  -.  ps )  ->  ( ph  /\  ps ) ) ) )
102, 9mpbird 166 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ph  \/  -.  ps )  \/  ( ph  /\  ps ) ) )
1110ex 114 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  \/  -.  ps )  \/  ( ph  /\  ps ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 682  DECID wdc 804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-stab 801  df-dc 805
This theorem is referenced by: (None)
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