Theorem ralcom3 2719

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 2610	. 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 2610	. 2 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑))
5	2, 4	impbii 126	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   ∈ wcel 2209  ∀wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502

This proof depends on definitions:  df-bi 117  df-ral 2533

This theorem is used by:  zfregfr  4723  tgss2  15305