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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 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))

Proof of Theorem notnotbdc
StepHypRef Expression
1 notnot 594 . 2 (𝜑 → ¬ ¬ 𝜑)
2 notnotrdc 789 . 2 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
31, 2impbid2 141 1 (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
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
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