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Theorem notnotbdc 777
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 569, 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 569 . 2 (𝜑 → ¬ ¬ 𝜑)
2 notnotrdc 762 . 2 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
31, 2impbid2 135 1 (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  con1biidc  782  imandc  797  imordc  807  dfbi3dc  1304  alexdc  1526
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