Theorem ss2ralv

Description: Two quantifications restricted to a subclass. (Contributed by AV, 11-Mar-2023)

	Ref	Expression
Assertion	ss2ralv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ssralv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑦 ∈ 𝐵 𝜑 → ∀ 𝑦 ∈ 𝐴 𝜑 ) )
2	1	ralimdv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐴 𝜑 ) )
3		ssralv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝜑 ) )
4	2 3	syld	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝜑 ) )