Theorem oibabs

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

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

Proof

Step	Hyp	Ref	Expression
1		norbi	⊢ ( ¬ ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ↔ 𝜓 ) )
2		id	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 ↔ 𝜓 ) )
3	1 2	ja	⊢ ( ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜑 ↔ 𝜓 ) )
4		ax-1	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ↔ 𝜓 ) ) )
5	3 4	impbii	⊢ ( ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ↔ 𝜓 ) ) ↔ ( 𝜑 ↔ 𝜓 ) )