Theorem ralv

Description: A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004)

	Ref	Expression
Assertion	ralv	⊢ ( ∀ 𝑥 ∈ V 𝜑 ↔ ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		df-ral	⊢ ( ∀ 𝑥 ∈ V 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ V → 𝜑 ) )
2		vex	⊢ 𝑥 ∈ V
3	2	a1bi	⊢ ( 𝜑 ↔ ( 𝑥 ∈ V → 𝜑 ) )
4	3	albii	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ V → 𝜑 ) )
5	1 4	bitr4i	⊢ ( ∀ 𝑥 ∈ V 𝜑 ↔ ∀ 𝑥 𝜑 )