Theorem reuv

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

	Ref	Expression
Assertion	reuv	$$⊢\exists !x\in \mathrm{V}\phi \leftrightarrow \exists !x\phi$$

Proof

Step	Hyp	Ref	Expression
1		df-reu	$$⊢\exists !x\in \mathrm{V}\phi \leftrightarrow \exists !x\left(x\in \mathrm{V}\wedge \phi \right)$$
2		vex	$$⊢x\in \mathrm{V}$$
3	2	biantrur	$$⊢\phi \leftrightarrow \left(x\in \mathrm{V}\wedge \phi \right)$$
4	3	eubii	$$⊢\exists !x\phi \leftrightarrow \exists !x\left(x\in \mathrm{V}\wedge \phi \right)$$
5	1 4	bitr4i	$$⊢\exists !x\in \mathrm{V}\phi \leftrightarrow \exists !x\phi$$