Theorem anabs1

Description: Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995) (Proof shortened by Wolf Lammen, 16-Nov-2013)

	Ref	Expression
Assertion	anabs1	$$⊢\left(\left(\phi \wedge \psi \right)\wedge \phi \right)\leftrightarrow \left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(\phi \wedge \psi \right)\to \phi$$
2	1	pm4.71i	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\wedge \phi \right)$$
3	2	bicomi	$$⊢\left(\left(\phi \wedge \psi \right)\wedge \phi \right)\leftrightarrow \left(\phi \wedge \psi \right)$$