Theorem reuimrmo

Description: Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo . (Contributed by Alexander van der Vekens, 25-Jun-2017)

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

Proof

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