Theorem rmoimia

Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017)

	Ref	Expression
Hypothesis	rmoimia.1	$$⊢x\in A\to \left(\phi \to \psi \right)$$
Assertion	rmoimia	$$⊢{\exists }^{*}x\in A\psi \to {\exists }^{*}x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		rmoimia.1	$$⊢x\in A\to \left(\phi \to \psi \right)$$
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	2 1	mprg	$$⊢{\exists }^{*}x\in A\psi \to {\exists }^{*}x\in A\phi$$