Theorem eu3v

Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994) Replace a nonfreeness hypothesis with a disjoint variable condition on ph , y to reduce axiom usage. (Revised by Wolf Lammen, 29-May-2019)

	Ref	Expression
Assertion	eu3v	$$⊢\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge \exists y\forall x\left(\phi \to x=y\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-eu	$$⊢\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge {\exists }^{*}x\phi \right)$$
2		dfmo	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$
3	2	anbi2i	$$⊢\left(\exists x\phi \wedge {\exists }^{*}x\phi \right)\leftrightarrow \left(\exists x\phi \wedge \exists y\forall x\left(\phi \to x=y\right)\right)$$
4	1 3	bitri	$$⊢\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge \exists y\forall x\left(\phi \to x=y\right)\right)$$