Theorem logic2 46026

Description: Variant of logic1 46024. (Contributed by Zhi Wang, 30-Aug-2024.)

Hypotheses

Ref	Expression
pm4.71da.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
logic2.2	⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)))

Assertion

Ref	Expression
logic2	⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))

Proof of Theorem logic2

Step	Hyp	Ref	Expression
1		pm4.71da.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	pm4.71da 46023	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))
3		logic2.2	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)))
4	2, 3	sylbid 239	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜏)))
5	1, 4	logic1 46024	1 ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395

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 206  df-an 396

This theorem is referenced by:  ralbidc  46034