Theorem imbi12d3

Description: Variant of imbi12d2 . (Contributed by Zhi Wang, 30-Aug-2024)

	Ref	Expression
Hypotheses	imbi12d2.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	imbi12d3.2	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\to \left(\theta \leftrightarrow \tau \right)\right)$$
Assertion	imbi12d3	$$⊢\phi \to \left(\left(\psi \to \theta \right)\leftrightarrow \left(\chi \to \tau \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		imbi12d2.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		imbi12d3.2	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\to \left(\theta \leftrightarrow \tau \right)\right)$$
3	1	pm4.71da	$$⊢\phi \to \left(\psi \leftrightarrow \left(\psi \wedge \chi \right)\right)$$
4	3 2	sylbid	$$⊢\phi \to \left(\psi \to \left(\theta \leftrightarrow \tau \right)\right)$$
5	1 4	imbi12d2	$$⊢\phi \to \left(\left(\psi \to \theta \right)\leftrightarrow \left(\chi \to \tau \right)\right)$$