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) (Revised by Gino Giotto, 10-Jan-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		nftru	$$⊢Ⅎy\top$$
4	1	a1i	$$⊢\top \to \underset{\_}{Ⅎ}xA$$
5	2	a1i	$$⊢\top \to Ⅎx\phi$$
6	3 4 5	nfraldw	$$⊢\top \to Ⅎx\forall y\in A\phi$$
7	6	mptru	$$⊢Ⅎx\forall y\in A\phi$$