Theorem exmoeu

Description: Existence is equivalent to uniqueness implying existential uniqueness. (Contributed by NM, 5-Apr-2004) (Proof shortened by Wolf Lammen, 5-Dec-2018) (Proof shortened by BJ, 7-Oct-2022)

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

Proof

Step	Hyp	Ref	Expression
1		exmoeub	$$⊢\exists x\phi \to \left({\exists }^{*}x\phi \leftrightarrow \exists !x\phi \right)$$
2	1	biimpd	$$⊢\exists x\phi \to \left({\exists }^{*}x\phi \to \exists !x\phi \right)$$
3		nexmo	$$⊢\neg \exists x\phi \to {\exists }^{*}x\phi$$
4	3	con1i	$$⊢\neg {\exists }^{*}x\phi \to \exists x\phi$$
5		euex	$$⊢\exists !x\phi \to \exists x\phi$$
6	4 5	ja	$$⊢\left({\exists }^{*}x\phi \to \exists !x\phi \right)\to \exists x\phi$$
7	2 6	impbii	$$⊢\exists x\phi \leftrightarrow \left({\exists }^{*}x\phi \to \exists !x\phi \right)$$