Theorem logic1a 46089

Description: Variant of logic1 46088. (Contributed by Zhi Wang, 30-Aug-2024.)

Hypotheses

Ref	Expression
pm4.71da.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
logic1a.2	⊢ ((𝜑 ∧ 𝜓) → (𝜃 ↔ 𝜏))

Assertion

Ref	Expression
logic1a	⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))

Proof of Theorem logic1a

Step	Hyp	Ref	Expression
1		pm4.71da.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		logic1a.2	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜃 ↔ 𝜏))
3	2	ex 412	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜏)))
4	1, 3	logic1 46088	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:  ralbidb  46097