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Theorem notnotdc 759
Description: Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot1 547, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.)
Assertion
Ref Expression
notnotdc (DECID φ → (φ ↔ ¬ ¬ φ))

Proof of Theorem notnotdc
StepHypRef Expression
1 notnot1 547 . 2 (φ → ¬ ¬ φ)
2 notnot2dc 742 . 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 733
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 532  ax-in2 533  ax-io 617
This theorem depends on definitions:  df-bi 110  df-dc 734
This theorem is referenced by:  con1biidc  764  imandc  779  imordc  789  dfbi3dc  1271  alexdc  1492
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