Theorem reueubd

Description: Restricted existential uniqueness is equivalent to existential uniqueness if the unique element is in the restricting class. (Contributed by AV, 4-Jan-2021)

	Ref	Expression
Hypothesis	reueubd.1	$$⊢\left(\phi \wedge \psi \right)\to x\in V$$
Assertion	reueubd	$$⊢\phi \to \left(\exists !x\in V\psi \leftrightarrow \exists !x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		reueubd.1	$$⊢\left(\phi \wedge \psi \right)\to x\in V$$
2		df-reu	$$⊢\exists !x\in V\psi \leftrightarrow \exists !x\left(x\in V\wedge \psi \right)$$
3	1	ex	$$⊢\phi \to \left(\psi \to x\in V\right)$$
4	3	pm4.71rd	$$⊢\phi \to \left(\psi \leftrightarrow \left(x\in V\wedge \psi \right)\right)$$
5	4	eubidv	$$⊢\phi \to \left(\exists !x\psi \leftrightarrow \exists !x\left(x\in V\wedge \psi \right)\right)$$
6	2 5	bitr4id	$$⊢\phi \to \left(\exists !x\in V\psi \leftrightarrow \exists !x\psi \right)$$