Theorem moim

Description: The at-most-one quantifier reverses implication. (Contributed by NM, 22-Apr-1995)

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

Proof

Step	Hyp	Ref	Expression
1		imim1	$$⊢\left(\phi \to \psi \right)\to \left(\left(\psi \to x=y\right)\to \left(\phi \to x=y\right)\right)$$
2	1	al2imi	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\psi \to x=y\right)\to \forall x\left(\phi \to x=y\right)\right)$$
3	2	eximdv	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists y\forall x\left(\psi \to x=y\right)\to \exists y\forall x\left(\phi \to x=y\right)\right)$$
4		dfmo	$$⊢{\exists }^{*}x\psi \leftrightarrow \exists y\forall x\left(\psi \to x=y\right)$$
5		dfmo	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$
6	3 4 5	3imtr4g	$$⊢\forall x\left(\phi \to \psi \right)\to \left({\exists }^{*}x\psi \to {\exists }^{*}x\phi \right)$$