Theorem ssralvOLD

Description: Obsolete version of ssralv as of 19-May-2025. (Contributed by NM, 11-Mar-2006) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	ssralvOLD	⊢ ( 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 ) )