Theorem notnotrdc 829

Description: Double negation elimination for a decidable proposition. The converse, notnot 619, holds for all propositions, not just decidable ones. This is Theorem *2.14 of [WhiteheadRussell] p. 102, but with a decidability condition added. (Contributed by Jim Kingdon, 11-Mar-2018.)

Assertion

Ref	Expression
notnotrdc	⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑))

Proof of Theorem notnotrdc

Step	Hyp	Ref	Expression
1		df-dc 821	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		orcom 718	. . 3 ⊢ ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ 𝜑))
3	1, 2	bitri 183	. 2 ⊢ (DECID 𝜑 ↔ (¬ 𝜑 ∨ 𝜑))
4		pm2.53 712	. 2 ⊢ ((¬ 𝜑 ∨ 𝜑) → (¬ ¬ 𝜑 → 𝜑))
5	3, 4	sylbi 120	1 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698  DECID wdc 820

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-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 821

This theorem is referenced by:  dcstab  830  notnotbdc  858  condandc  867  pm2.13dc  871  pm2.54dc  877  mkvprop  7040