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	$$⊢\left(\left(\phi \vee \psi \right)\to \left(\phi \leftrightarrow \psi \right)\right)\leftrightarrow \left(\phi \leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		norbi	$$⊢\neg \left(\phi \vee \psi \right)\to \left(\phi \leftrightarrow \psi \right)$$
2		id	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\phi \leftrightarrow \psi \right)$$
3	1 2	ja	$$⊢\left(\left(\phi \vee \psi \right)\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(\phi \leftrightarrow \psi \right)$$
4		ax-1	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\left(\phi \vee \psi \right)\to \left(\phi \leftrightarrow \psi \right)\right)$$
5	3 4	impbii	$$⊢\left(\left(\phi \vee \psi \right)\to \left(\phi \leftrightarrow \psi \right)\right)\leftrightarrow \left(\phi \leftrightarrow \psi \right)$$