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Theorem notnotrdc 844
Description: Double negation elimination for a decidable proposition. The converse, notnot 630, holds for all propositions, not just decidable ones. This is Theorem *2.14 of [WhiteheadRussell] p. 102, but with a decidability condition added. (Contributed by Jim Kingdon, 11-Mar-2018.)
Assertion
Ref Expression
notnotrdc  |-  (DECID  ph  ->  ( -.  -.  ph  ->  ph ) )

Proof of Theorem notnotrdc
StepHypRef Expression
1 df-dc 836 . . 3  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 orcom 729 . . 3  |-  ( (
ph  \/  -.  ph )  <->  ( -.  ph  \/  ph )
)
31, 2bitri 184 . 2  |-  (DECID  ph  <->  ( -.  ph  \/  ph ) )
4 pm2.53 723 . 2  |-  ( ( -.  ph  \/  ph )  ->  ( -.  -.  ph  ->  ph ) )
53, 4sylbi 121 1  |-  (DECID  ph  ->  ( -.  -.  ph  ->  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 709  DECID wdc 835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710
This theorem depends on definitions:  df-bi 117  df-dc 836
This theorem is referenced by:  dcstab  845  notnotbdc  873  condandc  882  pm2.13dc  886  pm2.54dc  892  mkvprop  7169  netap  7266
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