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Theorem imordc 867
Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 695, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.)
Assertion
Ref Expression
imordc  |-  (DECID  ph  ->  ( ( ph  ->  ps ) 
<->  ( -.  ph  \/  ps ) ) )

Proof of Theorem imordc
StepHypRef Expression
1 notnotbdc 842 . . 3  |-  (DECID  ph  ->  (
ph 
<->  -.  -.  ph )
)
21imbi1d 230 . 2  |-  (DECID  ph  ->  ( ( ph  ->  ps ) 
<->  ( -.  -.  ph  ->  ps ) ) )
3 dcn 812 . . 3  |-  (DECID  ph  -> DECID  -.  ph )
4 dfordc 862 . . 3  |-  (DECID  -.  ph  ->  ( ( -.  ph  \/  ps )  <->  ( -.  -.  ph  ->  ps )
) )
53, 4syl 14 . 2  |-  (DECID  ph  ->  ( ( -.  ph  \/  ps )  <->  ( -.  -.  ph 
->  ps ) ) )
62, 5bitr4d 190 1  |-  (DECID  ph  ->  ( ( ph  ->  ps ) 
<->  ( -.  ph  \/  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> 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:  pm4.62dc  868  pm2.26dc  877  nf4dc  1633  algcvgblem  11657  divgcdodd  11748
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