Theorem euf

Description: Version of eu6 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 12-Aug-1993) (Proof shortened by Wolf Lammen, 30-Oct-2018) Avoid ax-13 . (Revised by Wolf Lammen, 16-Oct-2022)

	Ref	Expression
Hypothesis	euf.1	$$⊢Ⅎy\phi$$
Assertion	euf	$$⊢\exists !x\phi \leftrightarrow \exists y\forall x\left(\phi \leftrightarrow x=y\right)$$

Proof

Step	Hyp	Ref	Expression
1		euf.1	$$⊢Ⅎy\phi$$
2		eu6	$$⊢\exists !x\phi \leftrightarrow \exists z\forall x\left(\phi \leftrightarrow x=z\right)$$
3		nfv	$$⊢Ⅎyx=z$$
4	1 3	nfbi	$$⊢Ⅎy\left(\phi \leftrightarrow x=z\right)$$
5	4	nfal	$$⊢Ⅎy\forall x\left(\phi \leftrightarrow x=z\right)$$
6		nfv	$$⊢Ⅎz\forall x\left(\phi \leftrightarrow x=y\right)$$
7		equequ2	$$⊢z=y\to \left(x=z\leftrightarrow x=y\right)$$
8	7	bibi2d	$$⊢z=y\to \left(\left(\phi \leftrightarrow x=z\right)\leftrightarrow \left(\phi \leftrightarrow x=y\right)\right)$$
9	8	albidv	$$⊢z=y\to \left(\forall x\left(\phi \leftrightarrow x=z\right)\leftrightarrow \forall x\left(\phi \leftrightarrow x=y\right)\right)$$
10	5 6 9	cbvexv1	$$⊢\exists z\forall x\left(\phi \leftrightarrow x=z\right)\leftrightarrow \exists y\forall x\left(\phi \leftrightarrow x=y\right)$$
11	2 10	bitri	$$⊢\exists !x\phi \leftrightarrow \exists y\forall x\left(\phi \leftrightarrow x=y\right)$$