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

Step	Hyp	Ref	Expression
1		notnot 601	. 2 ⊢ (𝜑 → ¬ ¬ 𝜑)
2		notnotrdc 811	. 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 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