Theorem ralcom3 2548

Description: A commutative law for restricted quantifiers that swaps the domain 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 82	. . 3 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)) → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝜑)))
2	1	ralimi2 2446	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) → ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))
3		pm2.04 82	. . 3 ⊢ ((𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝜑)) → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)))
4	3	ralimi2 2446	. 2 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑))
5	2, 4	impbii 125	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 1445  ∀wral 2370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390

This theorem depends on definitions:  df-bi 116  df-ral 2375

This theorem is referenced by:  zfregfr  4417  tgss2  11946