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	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	logic1.2	⊢ ( 𝜑 → ( 𝜓 → ( 𝜃 ↔ 𝜏 ) ) )
Assertion	logic1	⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜒 → 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm4.71da.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		logic1.2	⊢ ( 𝜑 → ( 𝜓 → ( 𝜃 ↔ 𝜏 ) ) )
3	2	pm5.74d	⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜓 → 𝜏 ) ) )
4	1	imbi1d	⊢ ( 𝜑 → ( ( 𝜓 → 𝜏 ) ↔ ( 𝜒 → 𝜏 ) ) )
5	3 4	bitrd	⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜒 → 𝜏 ) ) )