Theorem pm2.68dc 879

Description: Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 737 and one half of dfor2dc 880. (Contributed by Jim Kingdon, 27-Mar-2018.)

Assertion

Ref	Expression
pm2.68dc	⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓)))

Proof of Theorem pm2.68dc

Step	Hyp	Ref	Expression
1		jarl 647	. 2 ⊢ (((𝜑 → 𝜓) → 𝜓) → (¬ 𝜑 → 𝜓))
2		pm2.54dc 876	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)))
3	1, 2	syl5 32	1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 697  DECID wdc 819

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-dc 820

This theorem is referenced by:  dfor2dc  880