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Theorem pm5.6dc 896
Description: Conjunction in antecedent versus disjunction in consequent, for a decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with decidability condition added. The reverse implication holds for all propositions (see pm5.6r 897). (Contributed by Jim Kingdon, 2-Apr-2018.)
Assertion
Ref Expression
pm5.6dc  |-  (DECID  ps  ->  ( ( ( ph  /\  -.  ps )  ->  ch ) 
<->  ( ph  ->  ( ps  \/  ch ) ) ) )

Proof of Theorem pm5.6dc
StepHypRef Expression
1 dfordc 862 . . 3  |-  (DECID  ps  ->  ( ( ps  \/  ch ) 
<->  ( -.  ps  ->  ch ) ) )
21imbi2d 229 . 2  |-  (DECID  ps  ->  ( ( ph  ->  ( ps  \/  ch ) )  <-> 
( ph  ->  ( -. 
ps  ->  ch ) ) ) )
3 impexp 261 . 2  |-  ( ( ( ph  /\  -.  ps )  ->  ch )  <->  (
ph  ->  ( -.  ps  ->  ch ) ) )
42, 3syl6rbbr 198 1  |-  (DECID  ps  ->  ( ( ( ph  /\  -.  ps )  ->  ch ) 
<->  ( ph  ->  ( ps  \/  ch ) ) ) )
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-dc 805
This theorem is referenced by: (None)
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