Theorem abai

Description: Introduce one conjunct as an antecedent to the other. "abai" stands for "and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993) (Proof shortened by Wolf Lammen, 7-Dec-2012)

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

Proof

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