Theorem alral

Description: Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006)

	Ref	Expression
Assertion	alral	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ala1	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
2		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
3	1 2	sylibr	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 )