Theorem pm2.54dc 892

Description: Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 723, 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 843	. 2 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
2		notnotrdc 844	. . . . 5 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑))
3		orc 713	. . . . 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 712	. . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
7	6	a1i 9	. . 3 ⊢ (DECID 𝜑 → (𝜓 → (𝜑 ∨ 𝜓)))
8	5, 7	jaddc 865	. 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 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-in1 615  ax-in2 616  ax-io 710

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

This theorem is referenced by:  dfordc  893  pm2.68dc  895  pm4.79dc  904  pm5.11dc  910