Theorem moaneu

Description: Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006) (Proof shortened by Wolf Lammen, 27-Dec-2018)

	Ref	Expression
Assertion	moaneu	$$⊢{\exists }^{*}x\left(\phi \wedge \exists !x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		moanmo	$$⊢{\exists }^{*}x\left(\phi \wedge {\exists }^{*}x\phi \right)$$
2		eumo	$$⊢\exists !x\phi \to {\exists }^{*}x\phi$$
3	2	anim2i	$$⊢\left(\phi \wedge \exists !x\phi \right)\to \left(\phi \wedge {\exists }^{*}x\phi \right)$$
4	3	moimi	$$⊢{\exists }^{*}x\left(\phi \wedge {\exists }^{*}x\phi \right)\to {\exists }^{*}x\left(\phi \wedge \exists !x\phi \right)$$
5	1 4	ax-mp	$$⊢{\exists }^{*}x\left(\phi \wedge \exists !x\phi \right)$$