Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  notnotbdc GIF version

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 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))

Proof of Theorem notnotbdc
StepHypRef Expression
1 notnot 601 . 2 (𝜑 → ¬ ¬ 𝜑)
2 notnotrdc 811 . 2 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
31, 2impbid2 142 1 (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 802 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 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
 Copyright terms: Public domain W3C validator