Theorem 2euswapv

Description: A condition allowing to swap an existential quantifier and a unique existential quantifier. Version of 2euswap with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 10-Apr-2004) (Revised by Gino Giotto, 22-Aug-2023)

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

Proof

Step	Hyp	Ref	Expression
1		excomim	$$⊢\exists x\exists y\phi \to \exists y\exists x\phi$$
2	1	a1i	$$⊢\forall x{\exists }^{*}y\phi \to \left(\exists x\exists y\phi \to \exists y\exists x\phi \right)$$
3		2moswapv	$$⊢\forall x{\exists }^{*}y\phi \to \left({\exists }^{*}x\exists y\phi \to {\exists }^{*}y\exists x\phi \right)$$
4	2 3	anim12d	$$⊢\forall x{\exists }^{*}y\phi \to \left(\left(\exists x\exists y\phi \wedge {\exists }^{*}x\exists y\phi \right)\to \left(\exists y\exists x\phi \wedge {\exists }^{*}y\exists x\phi \right)\right)$$
5		df-eu	$$⊢\exists !x\exists y\phi \leftrightarrow \left(\exists x\exists y\phi \wedge {\exists }^{*}x\exists y\phi \right)$$
6		df-eu	$$⊢\exists !y\exists x\phi \leftrightarrow \left(\exists y\exists x\phi \wedge {\exists }^{*}y\exists x\phi \right)$$
7	4 5 6	3imtr4g	$$⊢\forall x{\exists }^{*}y\phi \to \left(\exists !x\exists y\phi \to \exists !y\exists x\phi \right)$$