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

Theorem notnotbdc 804
Description: Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 594, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.)
Assertion
Ref Expression
notnotbdc  |-  (DECID  ph  ->  (
ph 
<->  -.  -.  ph )
)

Proof of Theorem notnotbdc
StepHypRef Expression
1 notnot 594 . 2  |-  ( ph  ->  -.  -.  ph )
2 notnotrdc 789 . 2  |-  (DECID  ph  ->  ( -.  -.  ph  ->  ph ) )
31, 2impbid2 141 1  |-  (DECID  ph  ->  (
ph 
<->  -.  -.  ph )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103  DECID wdc 780
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 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by:  con1biidc  809  imandc  824  imordc  834  dfbi3dc  1333  alexdc  1555
  Copyright terms: Public domain W3C validator