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Theorem imandc 820
Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 819, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.)
Assertion
Ref Expression
imandc  |-  (DECID  ps  ->  ( ( ph  ->  ps ) 
<->  -.  ( ph  /\  -.  ps ) ) )

Proof of Theorem imandc
StepHypRef Expression
1 notnotbdc 800 . . 3  |-  (DECID  ps  ->  ( ps  <->  -.  -.  ps )
)
21imbi2d 228 . 2  |-  (DECID  ps  ->  ( ( ph  ->  ps ) 
<->  ( ph  ->  -.  -.  ps ) ) )
3 imnan 657 . 2  |-  ( (
ph  ->  -.  -.  ps )  <->  -.  ( ph  /\  -.  ps ) )
42, 3syl6bb 194 1  |-  (DECID  ps  ->  ( ( ph  ->  ps ) 
<->  -.  ( ph  /\  -.  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103  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:  annimdc  879  isprm3  10644
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