Theorem ssralv

Description: Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006)

	Ref	Expression
Assertion	ssralv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ssel	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
2	1	imim1d	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑥 ∈ 𝐵 → 𝜑 ) → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
3	2	ralimdv2	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 ) )