Theorem imbi12d2

Description: Distribution of implication over biconditional with replacement (deduction form). (Contributed by Zhi Wang, 30-Aug-2024)

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

Proof

Step	Hyp	Ref	Expression
1		imbi12d2.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		imbi12d2.2	$$⊢\phi \to \left(\psi \to \left(\theta \leftrightarrow \tau \right)\right)$$
3	2	pm5.74d	$$⊢\phi \to \left(\left(\psi \to \theta \right)\leftrightarrow \left(\psi \to \tau \right)\right)$$
4	1	imbi1d	$$⊢\phi \to \left(\left(\psi \to \tau \right)\leftrightarrow \left(\chi \to \tau \right)\right)$$
5	3 4	bitrd	$$⊢\phi \to \left(\left(\psi \to \theta \right)\leftrightarrow \left(\chi \to \tau \right)\right)$$