Theorem pm5.32d

Description: Distribution of implication over biconditional (deduction form). (Contributed by NM, 29-Oct-1996)

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

Proof

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