Theorem pm2.54dc 881

Description: Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 712, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.)

Assertion

Ref	Expression
pm2.54dc	⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)))

Proof of Theorem pm2.54dc

Step	Hyp	Ref	Expression
1		dcn 832	. 2 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
2		notnotrdc 833	. . . . 5 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑))
3		orc 702	. . . . 5 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
4	2, 3	syl6 33	. . . 4 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → (𝜑 ∨ 𝜓)))
5	4	a1d 22	. . 3 ⊢ (DECID 𝜑 → (DECID ¬ 𝜑 → (¬ ¬ 𝜑 → (𝜑 ∨ 𝜓))))
6		olc 701	. . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
7	6	a1i 9	. . 3 ⊢ (DECID 𝜑 → (𝜓 → (𝜑 ∨ 𝜓)))
8	5, 7	jaddc 854	. 2 ⊢ (DECID 𝜑 → (DECID ¬ 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓))))
9	1, 8	mpd 13	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698  DECID wdc 824

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-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 825

This theorem is referenced by:  dfordc  882  pm2.68dc  884  pm4.79dc  893  pm5.11dc  899