Theorem dfor2dc 896

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		pm2.62 749	. 2 ⊢ ((𝜑 ∨ 𝜓) → ((𝜑 → 𝜓) → 𝜓))
2		pm2.68dc 895	. 2 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓)))
3	1, 2	impbid2 143	1 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ ((𝜑 → 𝜓) → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 105   ∨ 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:  imimorbdc  897