Theorem dfmoeu

Description: An elementary proof of moeu in disguise, connecting an expression characterizing uniqueness ( df-mo ) to that of existential uniqueness ( eu6 ). No particular order of definition is required, as one can be derived from the other. This is shown here and in dfeumo . (Contributed by Wolf Lammen, 27-May-2019)

	Ref	Expression
Assertion	dfmoeu	$$⊢\left(\exists x\phi \to \exists y\forall x\left(\phi \leftrightarrow x=y\right)\right)\leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$

Proof

Step	Hyp	Ref	Expression
1		alnex	$$⊢\forall x\neg \phi \leftrightarrow \neg \exists x\phi$$
2		pm2.21	$$⊢\neg \phi \to \left(\phi \to x=y\right)$$
3	2	alimi	$$⊢\forall x\neg \phi \to \forall x\left(\phi \to x=y\right)$$
4	1 3	sylbir	$$⊢\neg \exists x\phi \to \forall x\left(\phi \to x=y\right)$$
5	4	19.8ad	$$⊢\neg \exists x\phi \to \exists y\forall x\left(\phi \to x=y\right)$$
6		biimp	$$⊢\left(\phi \leftrightarrow x=y\right)\to \left(\phi \to x=y\right)$$
7	6	alimi	$$⊢\forall x\left(\phi \leftrightarrow x=y\right)\to \forall x\left(\phi \to x=y\right)$$
8	7	eximi	$$⊢\exists y\forall x\left(\phi \leftrightarrow x=y\right)\to \exists y\forall x\left(\phi \to x=y\right)$$
9	5 8	ja	$$⊢\left(\exists x\phi \to \exists y\forall x\left(\phi \leftrightarrow x=y\right)\right)\to \exists y\forall x\left(\phi \to x=y\right)$$
10		nfia1	$$⊢Ⅎx\left(\forall x\left(\phi \to x=y\right)\to \forall x\left(\phi \leftrightarrow x=y\right)\right)$$
11		id	$$⊢\phi \to \phi$$
12		ax12v	$$⊢x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$
13	12	com12	$$⊢\phi \to \left(x=y\to \forall x\left(x=y\to \phi \right)\right)$$
14	11 13	embantd	$$⊢\phi \to \left(\left(\phi \to x=y\right)\to \forall x\left(x=y\to \phi \right)\right)$$
15	14	spsd	$$⊢\phi \to \left(\forall x\left(\phi \to x=y\right)\to \forall x\left(x=y\to \phi \right)\right)$$
16	15	ancld	$$⊢\phi \to \left(\forall x\left(\phi \to x=y\right)\to \left(\forall x\left(\phi \to x=y\right)\wedge \forall x\left(x=y\to \phi \right)\right)\right)$$
17		albiim	$$⊢\forall x\left(\phi \leftrightarrow x=y\right)\leftrightarrow \left(\forall x\left(\phi \to x=y\right)\wedge \forall x\left(x=y\to \phi \right)\right)$$
18	16 17	imbitrrdi	$$⊢\phi \to \left(\forall x\left(\phi \to x=y\right)\to \forall x\left(\phi \leftrightarrow x=y\right)\right)$$
19	10 18	exlimi	$$⊢\exists x\phi \to \left(\forall x\left(\phi \to x=y\right)\to \forall x\left(\phi \leftrightarrow x=y\right)\right)$$
20	19	eximdv	$$⊢\exists x\phi \to \left(\exists y\forall x\left(\phi \to x=y\right)\to \exists y\forall x\left(\phi \leftrightarrow x=y\right)\right)$$
21	20	com12	$$⊢\exists y\forall x\left(\phi \to x=y\right)\to \left(\exists x\phi \to \exists y\forall x\left(\phi \leftrightarrow x=y\right)\right)$$
22	9 21	impbii	$$⊢\left(\exists x\phi \to \exists y\forall x\left(\phi \leftrightarrow x=y\right)\right)\leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$