Theorem moexex

Description: "At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 . Use the version moexexvw when possible. (Contributed by NM, 3-Dec-2001) (Proof shortened by Wolf Lammen, 28-Dec-2018) Factor out common proof lines with moexexvw . (Revised by Wolf Lammen, 2-Oct-2023) (New usage is discouraged.)

	Ref	Expression
Hypothesis	moexex.1	$$⊢Ⅎy\phi$$
Assertion	moexex	$$⊢\left({\exists }^{*}x\phi \wedge \forall x{\exists }^{*}y\psi \right)\to {\exists }^{*}y\exists x\left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		moexex.1	$$⊢Ⅎy\phi$$
2	1	nfmo	$$⊢Ⅎy{\exists }^{*}x\phi$$
3		nfe1	$$⊢Ⅎx\exists x\left(\phi \wedge \psi \right)$$
4	3	nfmo	$$⊢Ⅎx{\exists }^{*}y\exists x\left(\phi \wedge \psi \right)$$
5	1 2 4	moexexlem	$$⊢\left({\exists }^{*}x\phi \wedge \forall x{\exists }^{*}y\psi \right)\to {\exists }^{*}y\exists x\left(\phi \wedge \psi \right)$$