Theorem ralbidb 45624

Description: Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc 45625 for a more generalized form. (Contributed by Zhi Wang, 6-Sep-2024.)

Hypotheses

Ref	Expression
ralbidb.1	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)))
ralbidb.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 ↔ 𝜃))

Assertion

Ref	Expression
ralbidb	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃)))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝜃(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem ralbidb

Step	Hyp	Ref	Expression
1		ralbidb.1	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)))
2		ralbidb.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 ↔ 𝜃))
3	1, 2	logic1a 45619	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜒) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜓) → 𝜃)))
4		impexp 454	. . 3 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜓) → 𝜃) ↔ (𝑥 ∈ 𝐵 → (𝜓 → 𝜃)))
5	3, 4	bitrdi 290	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜒) ↔ (𝑥 ∈ 𝐵 → (𝜓 → 𝜃))))
6	5	ralbidv2 3124	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111  ∀wral 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911

This theorem depends on definitions:  df-bi 210  df-an 400  df-ral 3075

This theorem is referenced by: (None)