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)

	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		nftru	$$⊢Ⅎy\top$$
5		nfcvd	$$⊢\top \to \underset{\_}{Ⅎ}xy$$
6	1	a1i	$$⊢\top \to \underset{\_}{Ⅎ}xA$$
7	5 6	nfeld	$$⊢\top \to Ⅎxy\in A$$
8	2	a1i	$$⊢\top \to Ⅎx\phi$$
9	7 8	nfand	$$⊢\top \to Ⅎx\left(y\in A\wedge \phi \right)$$
10	4 9	nfeudw	$$⊢\top \to Ⅎx\exists !y\left(y\in A\wedge \phi \right)$$
11	3 10	nfxfrd	$$⊢\top \to Ⅎx\exists !y\in A\phi$$
12	11	mptru	$$⊢Ⅎx\exists !y\in A\phi$$