Theorem logic1 48669

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

Hypotheses

Ref	Expression
pm4.71da.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
logic1.2	⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜏)))

Assertion

Ref	Expression
logic1	⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))

Proof of Theorem logic1

Step	Hyp	Ref	Expression
1		logic1.2	. . 3 ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜏)))
2	1	pm5.74d 273	. 2 ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜓 → 𝜏)))
3		pm4.71da.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	3	imbi1d 341	. 2 ⊢ (𝜑 → ((𝜓 → 𝜏) ↔ (𝜒 → 𝜏)))
5	2, 4	bitrd 279	1 ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))

Syntax hints:   → wi 4   ↔ wb 206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207

This theorem is referenced by:  logic1a  48670  logic2  48671