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Theorem pm3.11dc 899
Description: Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 704, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.)
Assertion
Ref Expression
pm3.11dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( -. 
ph  \/  -.  ps )  ->  ( ph  /\  ps ) ) ) )

Proof of Theorem pm3.11dc
StepHypRef Expression
1 anordc 898 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  /\  ps )  <->  -.  ( -.  ph  \/  -.  ps )
) ) )
21imp 122 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  /\ 
ps )  <->  -.  ( -.  ph  \/  -.  ps ) ) )
32biimprd 156 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( -.  ph  \/  -.  ps )  ->  ( ph  /\  ps ) ) )
43ex 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:  pm3.12dc  900  pm3.13dc  901
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