Theorem logic1

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

	Ref	Expression
Hypotheses	pm4.71da.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	logic1.2	$$⊢\phi \to \left(\psi \to \left(\theta \leftrightarrow \tau \right)\right)$$
Assertion	logic1	$$⊢\phi \to \left(\left(\psi \to \theta \right)\leftrightarrow \left(\chi \to \tau \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pm4.71da.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		logic1.2	$$⊢\phi \to \left(\psi \to \left(\theta \leftrightarrow \tau \right)\right)$$
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	$$⊢\phi \to \left(\left(\psi \to \theta \right)\leftrightarrow \left(\chi \to \tau \right)\right)$$