Theorem imbi12d3 49724

Description: Variant of imbi12d2 49722. (Contributed by Zhi Wang, 30-Aug-2024.)

Hypotheses

Ref	Expression
imbi12d2.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
imbi12d3.2	⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)))

Assertion

Ref	Expression
imbi12d3	⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))

Proof of Theorem imbi12d3

Step	Hyp	Ref	Expression
1		imbi12d2.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	pm4.71da 573	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))
3		imbi12d3.2	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)))
4	2, 3	sylbid 243	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜏)))
5	1, 4	imbi12d2 49722	1 ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  ralbidc  49732