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	|- F/_ x A
	nfralw.2	|- F/ x ph
Assertion	nfralw	|- F/ x A. y e. A ph

Proof

Step	Hyp	Ref	Expression
1		nfralw.1	|- F/_ x A
2		nfralw.2	|- F/ x ph
3	1	nfcri	|- F/ x y e. A
4	3	nf5ri	|- ( y e. A -> A. x y e. A )
5	2	nf5ri	|- ( ph -> A. x ph )
6	4 5	hbral	|- ( A. y e. A ph -> A. x A. y e. A ph )
7	6	nf5i	|- F/ x A. y e. A ph