Theorem rmoim

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

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

Proof

Step	Hyp	Ref	Expression
1		df-ral	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \forall x\left(x\in A\to \left(\phi \to \psi \right)\right)$$
2		imdistan	$$⊢\left(x\in A\to \left(\phi \to \psi \right)\right)\leftrightarrow \left(\left(x\in A\wedge \phi \right)\to \left(x\in A\wedge \psi \right)\right)$$
3	2	albii	$$⊢\forall x\left(x\in A\to \left(\phi \to \psi \right)\right)\leftrightarrow \forall x\left(\left(x\in A\wedge \phi \right)\to \left(x\in A\wedge \psi \right)\right)$$
4	1 3	bitri	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \forall x\left(\left(x\in A\wedge \phi \right)\to \left(x\in A\wedge \psi \right)\right)$$
5		moim	$$⊢\forall x\left(\left(x\in A\wedge \phi \right)\to \left(x\in A\wedge \psi \right)\right)\to \left({\exists }^{*}x\left(x\in A\wedge \psi \right)\to {\exists }^{*}x\left(x\in A\wedge \phi \right)\right)$$
6		df-rmo	$$⊢{\exists }^{*}x\in A\psi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \psi \right)$$
7		df-rmo	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \phi \right)$$
8	5 6 7	3imtr4g	$$⊢\forall x\left(\left(x\in A\wedge \phi \right)\to \left(x\in A\wedge \psi \right)\right)\to \left({\exists }^{*}x\in A\psi \to {\exists }^{*}x\in A\phi \right)$$
9	4 8	sylbi	$$⊢\forall x\in A\left(\phi \to \psi \right)\to \left({\exists }^{*}x\in A\psi \to {\exists }^{*}x\in A\phi \right)$$