Theorem pm2.68dc 793

Description: Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 667 and one half of dfor2dc 794. (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 584	. 2 ⊢ (((𝜑 → 𝜓) → 𝜓) → (¬ 𝜑 → 𝜓))
2		pm2.54dc 790	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)))
3	1, 2	syl5 28	1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓)))

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-in1 544  ax-in2 545  ax-io 630

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

This theorem is referenced by:  dfor2dc  794