Theorem reuv

Description: A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010)

	Ref	Expression
Assertion	reuv	⊢ ( ∃! 𝑥 ∈ V 𝜑 ↔ ∃! 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		df-reu	⊢ ( ∃! 𝑥 ∈ V 𝜑 ↔ ∃! 𝑥 ( 𝑥 ∈ V ∧ 𝜑 ) )
2		vex	⊢ 𝑥 ∈ V
3	2	biantrur	⊢ ( 𝜑 ↔ ( 𝑥 ∈ V ∧ 𝜑 ) )
4	3	eubii	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑥 ( 𝑥 ∈ V ∧ 𝜑 ) )
5	1 4	bitr4i	⊢ ( ∃! 𝑥 ∈ V 𝜑 ↔ ∃! 𝑥 𝜑 )