Theorem ralcom3 3266

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

Assertion

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

Proof of Theorem ralcom3

Step	Hyp	Ref	Expression
1		pm2.04 90	. . 3 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)) → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝜑)))
2	1	ralimi2 3072	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) → ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))
3		pm2.04 90	. . 3 ⊢ ((𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝜑)) → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)))
4	3	ralimi2 3072	. 2 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑))
5	2, 4	impbii 212	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2113  ∀wral 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816

This theorem depends on definitions:  df-bi 210  df-ral 3058

This theorem is referenced by:  tgss2  21731  ist1-3  22093  isreg2  22121