Theorem 2eu7

Description: Two equivalent expressions for double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 19-Feb-2005) (New usage is discouraged.)

	Ref	Expression
Assertion	2eu7	$$⊢\left(\exists !x\exists y\phi \wedge \exists !y\exists x\phi \right)\leftrightarrow \exists !x\exists !y\left(\exists x\phi \wedge \exists y\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfe1	$$⊢Ⅎx\exists x\phi$$
2	1	nfeu	$$⊢Ⅎx\exists !y\exists x\phi$$
3	2	euan	$$⊢\exists !x\left(\exists !y\exists x\phi \wedge \exists y\phi \right)\leftrightarrow \left(\exists !y\exists x\phi \wedge \exists !x\exists y\phi \right)$$
4		ancom	$$⊢\left(\exists x\phi \wedge \exists y\phi \right)\leftrightarrow \left(\exists y\phi \wedge \exists x\phi \right)$$
5	4	eubii	$$⊢\exists !y\left(\exists x\phi \wedge \exists y\phi \right)\leftrightarrow \exists !y\left(\exists y\phi \wedge \exists x\phi \right)$$
6		nfe1	$$⊢Ⅎy\exists y\phi$$
7	6	euan	$$⊢\exists !y\left(\exists y\phi \wedge \exists x\phi \right)\leftrightarrow \left(\exists y\phi \wedge \exists !y\exists x\phi \right)$$
8		ancom	$$⊢\left(\exists y\phi \wedge \exists !y\exists x\phi \right)\leftrightarrow \left(\exists !y\exists x\phi \wedge \exists y\phi \right)$$
9	5 7 8	3bitri	$$⊢\exists !y\left(\exists x\phi \wedge \exists y\phi \right)\leftrightarrow \left(\exists !y\exists x\phi \wedge \exists y\phi \right)$$
10	9	eubii	$$⊢\exists !x\exists !y\left(\exists x\phi \wedge \exists y\phi \right)\leftrightarrow \exists !x\left(\exists !y\exists x\phi \wedge \exists y\phi \right)$$
11		ancom	$$⊢\left(\exists !x\exists y\phi \wedge \exists !y\exists x\phi \right)\leftrightarrow \left(\exists !y\exists x\phi \wedge \exists !x\exists y\phi \right)$$
12	3 10 11	3bitr4ri	$$⊢\left(\exists !x\exists y\phi \wedge \exists !y\exists x\phi \right)\leftrightarrow \exists !x\exists !y\left(\exists x\phi \wedge \exists y\phi \right)$$