Theorem notnotrdc 844

Description: Double negation elimination for a decidable proposition. The converse, notnot 630, 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 836	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		orcom 729	. . 3 ⊢ ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ 𝜑))
3	1, 2	bitri 184	. 2 ⊢ (DECID 𝜑 ↔ (¬ 𝜑 ∨ 𝜑))
4		pm2.53 723	. 2 ⊢ ((¬ 𝜑 ∨ 𝜑) → (¬ ¬ 𝜑 → 𝜑))
5	3, 4	sylbi 121	1 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 709  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-in2 616  ax-io 710

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

This theorem is referenced by:  dcstab  845  notnotbdc  873  condandc  882  pm2.13dc  886  pm2.54dc  892  mkvprop  7224  netap  7321  bitsfzo  12119