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

Theorem notnotbdc 872
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 629, 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 629 . 2 (𝜑 → ¬ ¬ 𝜑)
2 notnotrdc 843 . 2 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
31, 2impbid2 143 1 (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  DECID wdc 834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-dc 835
This theorem is referenced by:  con1biidc  877  imordc  897  dfbi3dc  1397  alexdc  1619
  Copyright terms: Public domain W3C validator