Theorem dfor2dc 885

Description: Disjunction expressed in terms of implication only, for a decidable proposition. Based on theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 27-Mar-2018.)

Assertion

Ref	Expression
dfor2dc	⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ ((𝜑 → 𝜓) → 𝜓)))

Proof of Theorem dfor2dc

Step	Hyp	Ref	Expression
1		pm2.62 738	. 2 ⊢ ((𝜑 ∨ 𝜓) → ((𝜑 → 𝜓) → 𝜓))
2		pm2.68dc 884	. 2 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓)))
3	1, 2	impbid2 142	1 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ ((𝜑 → 𝜓) → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 104   ∨ 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:  imimorbdc  886