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Theorem imordc 796
Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 797, 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 766 . . 3  |-  (DECID  ph  ->  (
ph 
<->  -.  -.  ph )
)
21imbi1d 220 . 2  |-  (DECID  ph  ->  ( ( ph  ->  ps ) 
<->  ( -.  -.  ph  ->  ps ) ) )
3 dcn 746 . . 3  |-  (DECID  ph  -> DECID  -.  ph )
4 dfordc 791 . . 3  |-  (DECID  -.  ph  ->  ( ( -.  ph  \/  ps )  <->  ( -.  -.  ph  ->  ps )
) )
53, 4syl 14 . 2  |-  (DECID  ph  ->  ( ( -.  ph  \/  ps )  <->  ( -.  -.  ph 
->  ps ) ) )
62, 5bitr4d 180 1  |-  (DECID  ph  ->  ( ( ph  ->  ps ) 
<->  ( -.  ph  \/  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 98    \/ wo 629  DECID wdc 742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630
This theorem depends on definitions:  df-bi 110  df-dc 743
This theorem is referenced by:  pm4.62dc  798  pm2.26dc  813  nf4dc  1560  algcvgblem  9888
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