Theorem nfralw

Description: Bound-variable hypothesis builder for restricted quantification. Version of nfral with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 1-Sep-1999) Avoid ax-13 . (Revised by Gino Giotto, 10-Jan-2024) (Proof shortened by Wolf Lammen, 13-Dec-2024)

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

Proof

Step	Hyp	Ref	Expression
1		nfralw.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfralw.2	$$⊢Ⅎx\phi$$
3	1	nfcri	$$⊢Ⅎxy\in A$$
4	3	nf5ri	$$⊢y\in A\to \forall xy\in A$$
5	2	nf5ri	$$⊢\phi \to \forall x\phi$$
6	4 5	hbral	$$⊢\forall y\in A\phi \to \forall x\forall y\in A\phi$$
7	6	nf5i	$$⊢Ⅎx\forall y\in A\phi$$