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Theorem notnotbdc 800
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 592, 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 592 . 2  |-  ( ph  ->  -.  -.  ph )
2 notnotrdc 785 . 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 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:  con1biidc  805  imandc  820  imordc  830  dfbi3dc  1329  alexdc  1551
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