Theorem ralcom3 3113

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		bi2.04 392	. 2 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝜑)))
2	1	ralbii2 3105	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2145  ∀wral 3077

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-ral 3078

This theorem is used by:  tgss2  23298  ist1-3  23660  isreg2  23688