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Theorem pm3.13dc 901
Description: Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 703, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
Assertion
Ref Expression
pm3.13dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  /\ 
ps )  ->  ( -.  ph  \/  -.  ps ) ) ) )

Proof of Theorem pm3.13dc
StepHypRef Expression
1 dcn 780 . . 3  |-  (DECID  ph  -> DECID  -.  ph )
2 dcn 780 . . 3  |-  (DECID  ps  -> DECID  -.  ps )
3 dcor 877 . . 3  |-  (DECID  -.  ph  ->  (DECID  -.  ps  -> DECID  ( -.  ph  \/  -.  ps ) ) )
41, 2, 3syl2im 38 . 2  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( -.  ph  \/  -.  ps ) ) )
5 pm3.11dc 899 . 2  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( -. 
ph  \/  -.  ps )  ->  ( ph  /\  ps ) ) ) )
6 con1dc 787 . 2  |-  (DECID  ( -. 
ph  \/  -.  ps )  ->  ( ( -.  ( -.  ph  \/  -.  ps )  ->  ( ph  /\  ps ) )  ->  ( -.  ( ph  /\  ps )  ->  ( -.  ph  \/  -.  ps ) ) ) )
74, 5, 6syl6c 65 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  /\ 
ps )  ->  ( -.  ph  \/  -.  ps ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ 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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