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Theorem notnotbdc 840
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 601, 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 601 . 2  |-  ( ph  ->  -.  -.  ph )
2 notnotrdc 811 . 2  |-  (DECID  ph  ->  ( -.  -.  ph  ->  ph ) )
31, 2impbid2 142 1  |-  (DECID  ph  ->  (
ph 
<->  -.  -.  ph )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104  DECID wdc 802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-dc 803
This theorem is referenced by:  con1biidc  845  imordc  865  dfbi3dc  1358  alexdc  1581
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