Theorem anabs5

Description: Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996) (Proof shortened by Wolf Lammen, 9-Dec-2012)

	Ref	Expression
Assertion	anabs5	$$⊢\left(\phi \wedge \left(\phi \wedge \psi \right)\right)\leftrightarrow \left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ibar	$$⊢\phi \to \left(\psi \leftrightarrow \left(\phi \wedge \psi \right)\right)$$
2	1	bicomd	$$⊢\phi \to \left(\left(\phi \wedge \psi \right)\leftrightarrow \psi \right)$$
3	2	pm5.32i	$$⊢\left(\phi \wedge \left(\phi \wedge \psi \right)\right)\leftrightarrow \left(\phi \wedge \psi \right)$$