ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  imordc Unicode version

Theorem imordc 830
Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 831, 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 800 . . 3  |-  (DECID  ph  ->  (
ph 
<->  -.  -.  ph )
)
21imbi1d 229 . 2  |-  (DECID  ph  ->  ( ( ph  ->  ps ) 
<->  ( -.  -.  ph  ->  ps ) ) )
3 dcn 780 . . 3  |-  (DECID  ph  -> DECID  -.  ph )
4 dfordc 825 . . 3  |-  (DECID  -.  ph  ->  ( ( -.  ph  \/  ps )  <->  ( -.  -.  ph  ->  ps )
) )
53, 4syl 14 . 2  |-  (DECID  ph  ->  ( ( -.  ph  \/  ps )  <->  ( -.  -.  ph 
->  ps ) ) )
62, 5bitr4d 189 1  |-  (DECID  ph  ->  ( ( ph  ->  ps ) 
<->  ( -.  ph  \/  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> 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:  pm4.62dc  832  pm2.26dc  847  nf4dc  1601  algcvgblem  10638  divgcdodd  10729
  Copyright terms: Public domain W3C validator