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Theorem notnotbdc 766
 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 559, 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 559 . 2 (𝜑 → ¬ ¬ 𝜑)
2 notnotrdc 751 . 2 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
31, 2impbid2 131 1 (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  DECID wdc 742 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  con1biidc  771  imandc  786  imordc  796  dfbi3dc  1288  alexdc  1510
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