Theorem ralcom3

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

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

Proof

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