Theorem nfrexw

Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 30-Dec-2019) Add disjoint variable condition to avoid ax-13 . See nfrex for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

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

Proof

Step	Hyp	Ref	Expression
1		nfralw.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfralw.2	$$⊢Ⅎx\phi$$
3		nftru	$$⊢Ⅎy\top$$
4	1	a1i	$$⊢\top \to \underset{\_}{Ⅎ}xA$$
5	2	a1i	$$⊢\top \to Ⅎx\phi$$
6	3 4 5	nfrexdw	$$⊢\top \to Ⅎx\exists y\in A\phi$$
7	6	mptru	$$⊢Ⅎx\exists y\in A\phi$$