Theorem 2eu8

Description: Two equivalent expressions for double existential uniqueness. Curiously, we can put E! on either of the internal conjuncts but not both. We can also commute E! x E! y using 2eu7 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 20-Feb-2005) (New usage is discouraged.)

	Ref	Expression
Assertion	2eu8	$$⊢\exists !x\exists !y\left(\exists x\phi \wedge \exists y\phi \right)\leftrightarrow \exists !x\exists !y\left(\exists !x\phi \wedge \exists y\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		2eu2	$$⊢\exists !x\exists y\phi \to \left(\exists !y\exists !x\phi \leftrightarrow \exists !y\exists x\phi \right)$$
2	1	pm5.32i	$$⊢\left(\exists !x\exists y\phi \wedge \exists !y\exists !x\phi \right)\leftrightarrow \left(\exists !x\exists y\phi \wedge \exists !y\exists x\phi \right)$$
3		nfeu1	$$⊢Ⅎx\exists !x\phi$$
4	3	nfeu	$$⊢Ⅎx\exists !y\exists !x\phi$$
5	4	euan	$$⊢\exists !x\left(\exists !y\exists !x\phi \wedge \exists y\phi \right)\leftrightarrow \left(\exists !y\exists !x\phi \wedge \exists !x\exists y\phi \right)$$
6		ancom	$$⊢\left(\exists !x\phi \wedge \exists y\phi \right)\leftrightarrow \left(\exists y\phi \wedge \exists !x\phi \right)$$
7	6	eubii	$$⊢\exists !y\left(\exists !x\phi \wedge \exists y\phi \right)\leftrightarrow \exists !y\left(\exists y\phi \wedge \exists !x\phi \right)$$
8		nfe1	$$⊢Ⅎy\exists y\phi$$
9	8	euan	$$⊢\exists !y\left(\exists y\phi \wedge \exists !x\phi \right)\leftrightarrow \left(\exists y\phi \wedge \exists !y\exists !x\phi \right)$$
10		ancom	$$⊢\left(\exists y\phi \wedge \exists !y\exists !x\phi \right)\leftrightarrow \left(\exists !y\exists !x\phi \wedge \exists y\phi \right)$$
11	7 9 10	3bitri	$$⊢\exists !y\left(\exists !x\phi \wedge \exists y\phi \right)\leftrightarrow \left(\exists !y\exists !x\phi \wedge \exists y\phi \right)$$
12	11	eubii	$$⊢\exists !x\exists !y\left(\exists !x\phi \wedge \exists y\phi \right)\leftrightarrow \exists !x\left(\exists !y\exists !x\phi \wedge \exists y\phi \right)$$
13		ancom	$$⊢\left(\exists !x\exists y\phi \wedge \exists !y\exists !x\phi \right)\leftrightarrow \left(\exists !y\exists !x\phi \wedge \exists !x\exists y\phi \right)$$
14	5 12 13	3bitr4ri	$$⊢\left(\exists !x\exists y\phi \wedge \exists !y\exists !x\phi \right)\leftrightarrow \exists !x\exists !y\left(\exists !x\phi \wedge \exists y\phi \right)$$
15		2eu7	$$⊢\left(\exists !x\exists y\phi \wedge \exists !y\exists x\phi \right)\leftrightarrow \exists !x\exists !y\left(\exists x\phi \wedge \exists y\phi \right)$$
16	2 14 15	3bitr3ri	$$⊢\exists !x\exists !y\left(\exists x\phi \wedge \exists y\phi \right)\leftrightarrow \exists !x\exists !y\left(\exists !x\phi \wedge \exists y\phi \right)$$