Theorem 2euexv

Description: Double quantification with existential uniqueness. Version of 2euex with x and y distinct, but not requiring ax-13 . (Contributed by NM, 3-Dec-2001) (Revised by Wolf Lammen, 2-Oct-2023)

	Ref	Expression
Assertion	2euexv	$$⊢\exists !x\exists y\phi \to \exists y\exists !x\phi$$

Proof

Step	Hyp	Ref	Expression
1		df-eu	$$⊢\exists !x\exists y\phi \leftrightarrow \left(\exists x\exists y\phi \wedge {\exists }^{*}x\exists y\phi \right)$$
2		excom	$$⊢\exists x\exists y\phi \leftrightarrow \exists y\exists x\phi$$
3		nfe1	$$⊢Ⅎy\exists y\phi$$
4	3	nfmov	$$⊢Ⅎy{\exists }^{*}x\exists y\phi$$
5		19.8a	$$⊢\phi \to \exists y\phi$$
6	5	moimi	$$⊢{\exists }^{*}x\exists y\phi \to {\exists }^{*}x\phi$$
7		moeu	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \to \exists !x\phi \right)$$
8	6 7	sylib	$$⊢{\exists }^{*}x\exists y\phi \to \left(\exists x\phi \to \exists !x\phi \right)$$
9	4 8	eximd	$$⊢{\exists }^{*}x\exists y\phi \to \left(\exists y\exists x\phi \to \exists y\exists !x\phi \right)$$
10	2 9	biimtrid	$$⊢{\exists }^{*}x\exists y\phi \to \left(\exists x\exists y\phi \to \exists y\exists !x\phi \right)$$
11	10	impcom	$$⊢\left(\exists x\exists y\phi \wedge {\exists }^{*}x\exists y\phi \right)\to \exists y\exists !x\phi$$
12	1 11	sylbi	$$⊢\exists !x\exists y\phi \to \exists y\exists !x\phi$$