Theorem pm5.32dav

Description: Distribution of implication over biconditional (deduction form). Variant of pm5.32da . (Contributed by Zhi Wang, 30-Aug-2024)

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

Proof

Step	Hyp	Ref	Expression
1		pm5.32dav.1	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \leftrightarrow \theta \right)$$
2	1	pm5.32da	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\leftrightarrow \left(\psi \wedge \theta \right)\right)$$
3		ancom	$$⊢\left(\psi \wedge \chi \right)\leftrightarrow \left(\chi \wedge \psi \right)$$
4		ancom	$$⊢\left(\psi \wedge \theta \right)\leftrightarrow \left(\theta \wedge \psi \right)$$
5	2 3 4	3bitr3g	$$⊢\phi \to \left(\left(\chi \wedge \psi \right)\leftrightarrow \left(\theta \wedge \psi \right)\right)$$