Theorem nfreuw

Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 30-Oct-2010) (Revised by Gino Giotto, 10-Jan-2024) Avoid ax-9 , ax-ext . (Revised by Wolf Lammen, 21-Nov-2024)

	Ref	Expression
Hypotheses	nfreuw.1	$$⊢\underset{\_}{Ⅎ}xA$$
	nfreuw.2	$$⊢Ⅎx\phi$$
Assertion	nfreuw	$$⊢Ⅎx\exists !y\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		nfreuw.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfreuw.2	$$⊢Ⅎx\phi$$
3		df-reu	$$⊢\exists !y\in A\phi \leftrightarrow \exists !y\left(y\in A\wedge \phi \right)$$
4	1	nfcri	$$⊢Ⅎxy\in A$$
5	4 2	nfan	$$⊢Ⅎx\left(y\in A\wedge \phi \right)$$
6	5	nfeuw	$$⊢Ⅎx\exists !y\left(y\in A\wedge \phi \right)$$
7	3 6	nfxfr	$$⊢Ⅎx\exists !y\in A\phi$$