Theorem oranabs

Description: Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton, 23-Jun-2005) (Proof shortened by Wolf Lammen, 10-Nov-2013)

	Ref	Expression
Assertion	oranabs	$$⊢\left(\left(\phi \vee \neg \psi \right)\wedge \psi \right)\leftrightarrow \left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		biortn	$$⊢\psi \to \left(\phi \leftrightarrow \left(\neg \psi \vee \phi \right)\right)$$
2		orcom	$$⊢\left(\neg \psi \vee \phi \right)\leftrightarrow \left(\phi \vee \neg \psi \right)$$
3	1 2	bitr2di	$$⊢\psi \to \left(\left(\phi \vee \neg \psi \right)\leftrightarrow \phi \right)$$
4	3	pm5.32ri	$$⊢\left(\left(\phi \vee \neg \psi \right)\wedge \psi \right)\leftrightarrow \left(\phi \wedge \psi \right)$$