Theorem oibabs 948

Description: Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)

Assertion

Ref	Expression
oibabs	⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) ↔ (𝜑 ↔ 𝜓))

Proof of Theorem oibabs

Step	Hyp	Ref	Expression
1		norbi 883	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓))
2		id 22	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
3	1, 2	ja 188	. 2 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) → (𝜑 ↔ 𝜓))
4		ax-1 6	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)))
5	3, 4	impbii 211	1 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) ↔ (𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∨ wo 843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-or 844

This theorem is referenced by: (None)