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Theorem anordc 898
Description: Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 704, holds for all propositions, but the equivalence only holds given decidability. (Contributed by Jim Kingdon, 21-Apr-2018.)
Assertion
Ref Expression
anordc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  /\  ps )  <->  -.  ( -.  ph  \/  -.  ps )
) ) )

Proof of Theorem anordc
StepHypRef Expression
1 dcan 876 . 2  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  /\  ps )
) )
2 ianordc 833 . . . . 5  |-  (DECID  ph  ->  ( -.  ( ph  /\  ps )  <->  ( -.  ph  \/  -.  ps ) ) )
32bicomd 139 . . . 4  |-  (DECID  ph  ->  ( ( -.  ph  \/  -.  ps )  <->  -.  ( ph  /\  ps ) ) )
43a1d 22 . . 3  |-  (DECID  ph  ->  (DECID  (
ph  /\  ps )  ->  ( ( -.  ph  \/  -.  ps )  <->  -.  ( ph  /\  ps ) ) ) )
54con2biddc 808 . 2  |-  (DECID  ph  ->  (DECID  (
ph  /\  ps )  ->  ( ( ph  /\  ps )  <->  -.  ( -.  ph  \/  -.  ps )
) ) )
61, 5syld 44 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:  pm3.11dc  899  dn1dc  902
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