Theorem notnotbdc 862

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 619, 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 619	. 2 ⊢ (𝜑 → ¬ ¬ 𝜑)
2		notnotrdc 833	. 2 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑))
3	1, 2	impbid2 142	1 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 825

This theorem is referenced by:  con1biidc  867  imordc  887  dfbi3dc  1387  alexdc  1607