Theorem pm2.1dc 745

Description: Commuted law of the excluded middle for a decidable proposition. Based on theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by Jim Kingdon, 25-Mar-2018.)

Assertion

Ref	Expression
pm2.1dc	⊢ (DECID 𝜑 → (¬ 𝜑 ∨ 𝜑))

Proof of Theorem pm2.1dc

Step	Hyp	Ref	Expression
1		df-dc 743	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		orcom 647	. . 3 ⊢ ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ 𝜑))
3	1, 2	bitri 173	. 2 ⊢ (DECID 𝜑 ↔ (¬ 𝜑 ∨ 𝜑))
4	3	biimpi 113	1 ⊢ (DECID 𝜑 → (¬ 𝜑 ∨ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  pm2.6dc  759  rabrsndc  3438