Theorem 2moswapv

Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Version of 2moswap with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 10-Apr-2004) (Revised by GG, 22-Aug-2023) Factor out common proof lines with moexexvw . (Revised by Wolf Lammen, 2-Oct-2023)

	Ref	Expression
Assertion	2moswapv	$$⊢\forall x{\exists }^{*}y\phi \to \left({\exists }^{*}x\exists y\phi \to {\exists }^{*}y\exists x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfe1	$$⊢Ⅎy\exists y\phi$$
2	1	nfmov	$$⊢Ⅎy{\exists }^{*}x\exists y\phi$$
3		nfe1	$$⊢Ⅎx\exists x\left(\exists y\phi \wedge \phi \right)$$
4	3	nfmov	$$⊢Ⅎx{\exists }^{*}y\exists x\left(\exists y\phi \wedge \phi \right)$$
5	1 2 4	moexexlem	$$⊢\left({\exists }^{*}x\exists y\phi \wedge \forall x{\exists }^{*}y\phi \right)\to {\exists }^{*}y\exists x\left(\exists y\phi \wedge \phi \right)$$
6	5	expcom	$$⊢\forall x{\exists }^{*}y\phi \to \left({\exists }^{*}x\exists y\phi \to {\exists }^{*}y\exists x\left(\exists y\phi \wedge \phi \right)\right)$$
7		19.8a	$$⊢\phi \to \exists y\phi$$
8	7	pm4.71ri	$$⊢\phi \leftrightarrow \left(\exists y\phi \wedge \phi \right)$$
9	8	exbii	$$⊢\exists x\phi \leftrightarrow \exists x\left(\exists y\phi \wedge \phi \right)$$
10	9	mobii	$$⊢{\exists }^{*}y\exists x\phi \leftrightarrow {\exists }^{*}y\exists x\left(\exists y\phi \wedge \phi \right)$$
11	6 10	imbitrrdi	$$⊢\forall x{\exists }^{*}y\phi \to \left({\exists }^{*}x\exists y\phi \to {\exists }^{*}y\exists x\phi \right)$$