Theorem notnotbdc 873

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 630, 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 630	. 2 ⊢ (𝜑 → ¬ ¬ 𝜑)
2		notnotrdc 844	. 2 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑))
3	1, 2	impbid2 143	1 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  DECID wdc 835

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 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-dc 836

This theorem is referenced by:  con1biidc  878  imordc  898  dfbi3dc  1408  alexdc  1633