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

Step	Hyp	Ref	Expression
1		notnot 559	. 2 ⊢ (𝜑 → ¬ ¬ 𝜑)
2		notnotrdc 751	. 2 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑))
3	1, 2	impbid2 131	1 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))

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