Theorem moeu

Description: Uniqueness is equivalent to existence implying unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by NM, 8-Mar-1995) This used to be the definition of the at-most-one quantifier, while df-mo was then proved as dfmo2 . (Revised by BJ, 30-Sep-2022)

	Ref	Expression
Assertion	moeu	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \to \exists !x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		moabs	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \to {\exists }^{*}x\phi \right)$$
2		exmoeub	$$⊢\exists x\phi \to \left({\exists }^{*}x\phi \leftrightarrow \exists !x\phi \right)$$
3	2	pm5.74i	$$⊢\left(\exists x\phi \to {\exists }^{*}x\phi \right)\leftrightarrow \left(\exists x\phi \to \exists !x\phi \right)$$
4	1 3	bitri	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \to \exists !x\phi \right)$$