Theorem moimi

Description: The at-most-one quantifier reverses implication. (Contributed by NM, 15-Feb-2006) Remove use of ax-5 . (Revised by Steven Nguyen, 9-May-2023)

	Ref	Expression
Hypothesis	moimi.1	$$⊢\phi \to \psi$$
Assertion	moimi	$$⊢{\exists }^{*}x\psi \to {\exists }^{*}x\phi$$

Proof

Step	Hyp	Ref	Expression
1		moimi.1	$$⊢\phi \to \psi$$
2	1	imim1i	$$⊢\left(\psi \to x=y\right)\to \left(\phi \to x=y\right)$$
3	2	alimi	$$⊢\forall x\left(\psi \to x=y\right)\to \forall x\left(\phi \to x=y\right)$$
4	3	eximi	$$⊢\exists y\forall x\left(\psi \to x=y\right)\to \exists y\forall x\left(\phi \to x=y\right)$$
5		df-mo	$$⊢{\exists }^{*}x\psi \leftrightarrow \exists y\forall x\left(\psi \to x=y\right)$$
6		df-mo	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$
7	4 5 6	3imtr4i	$$⊢{\exists }^{*}x\psi \to {\exists }^{*}x\phi$$