Theorem 2euswap

Description: A condition allowing to swap an existential quantifier and a unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker 2euswapv when possible. (Contributed by NM, 10-Apr-2004) (New usage is discouraged.)

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

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		2moswap	$$⊢\forall x{\exists }^{*}y\phi \to \left({\exists }^{*}x\exists y\phi \to {\exists }^{*}y\exists x\phi \right)$$
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	$$⊢\forall x{\exists }^{*}y\phi \to \left(\exists !x\exists y\phi \to \exists !y\exists x\phi \right)$$