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Theorem pm5.6dc 869
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 870). (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 825 . . 3  |-  (DECID  ps  ->  ( ( ps  \/  ch ) 
<->  ( -.  ps  ->  ch ) ) )
21imbi2d 228 . 2  |-  (DECID  ps  ->  ( ( ph  ->  ( ps  \/  ch ) )  <-> 
( ph  ->  ( -. 
ps  ->  ch ) ) ) )
3 impexp 259 . 2  |-  ( ( ( ph  /\  -.  ps )  ->  ch )  <->  (
ph  ->  ( -.  ps  ->  ch ) ) )
42, 3syl6rbbr 197 1  |-  (DECID  ps  ->  ( ( ( ph  /\  -.  ps )  ->  ch ) 
<->  ( ph  ->  ( ps  \/  ch ) ) ) )
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: (None)
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