Theorem pm5.32da

Description: Distribution of implication over biconditional (deduction form). (Contributed by NM, 9-Dec-2006)

	Ref	Expression
Hypothesis	pm5.32da.1	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \leftrightarrow \theta \right)$$
Assertion	pm5.32da	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\leftrightarrow \left(\psi \wedge \theta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pm5.32da.1	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \leftrightarrow \theta \right)$$
2	1	ex	$$⊢\phi \to \left(\psi \to \left(\chi \leftrightarrow \theta \right)\right)$$
3	2	pm5.32d	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\leftrightarrow \left(\psi \wedge \theta \right)\right)$$