Theorem rmoimi2

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

	Ref	Expression
Hypothesis	rmoimi2.1	$$⊢\forall x\left(\left(x\in A\wedge \phi \right)\to \left(x\in B\wedge \psi \right)\right)$$
Assertion	rmoimi2	$$⊢{\exists }^{*}x\in B\psi \to {\exists }^{*}x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		rmoimi2.1	$$⊢\forall x\left(\left(x\in A\wedge \phi \right)\to \left(x\in B\wedge \psi \right)\right)$$
2		moim	$$⊢\forall x\left(\left(x\in A\wedge \phi \right)\to \left(x\in B\wedge \psi \right)\right)\to \left({\exists }^{*}x\left(x\in B\wedge \psi \right)\to {\exists }^{*}x\left(x\in A\wedge \phi \right)\right)$$
3	1 2	ax-mp	$$⊢{\exists }^{*}x\left(x\in B\wedge \psi \right)\to {\exists }^{*}x\left(x\in A\wedge \phi \right)$$
4		df-rmo	$$⊢{\exists }^{*}x\in B\psi \leftrightarrow {\exists }^{*}x\left(x\in B\wedge \psi \right)$$
5		df-rmo	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \phi \right)$$
6	3 4 5	3imtr4i	$$⊢{\exists }^{*}x\in B\psi \to {\exists }^{*}x\in A\phi$$