Theorem euimmo

Description: Existential uniqueness implies uniqueness through reverse implication. (Contributed by NM, 22-Apr-1995)

	Ref	Expression
Assertion	euimmo	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists !x\psi \to {\exists }^{*}x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		eumo	$$⊢\exists !x\psi \to {\exists }^{*}x\psi$$
2		moim	$$⊢\forall x\left(\phi \to \psi \right)\to \left({\exists }^{*}x\psi \to {\exists }^{*}x\phi \right)$$
3	1 2	syl5	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists !x\psi \to {\exists }^{*}x\phi \right)$$