Theorem ralcom3 3098

Description: A commutation law for restricted universal quantifiers that swaps the domains of the restriction. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Wolf Lammen, 22-Dec-2024.)

Assertion

Ref	Expression
ralcom3	⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))

Proof of Theorem ralcom3

Step	Hyp	Ref	Expression
1		bi2.04 389	. 2 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝜑)))
2	1	ralbii2 3090	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2106  ∀wral 3062

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

This theorem depends on definitions:  df-bi 206  df-ral 3063

This theorem is referenced by:  tgss2  22250  ist1-3  22613  isreg2  22641