Theorem oibabs 966

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 900	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓))
2		id 23	. . 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 212	1 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) ↔ (𝜑 ↔ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 862

This theorem is used by: (None)