Theorem nfrals

Description: Bound-variable hypothesis builder for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypotheses	nfrals.1	|- F/_ x A
	nfrals.2	|- F/ x ph
	nfrals.3	|- F/ x ps
Assertion	nfrals	|- F/ x AE y e. A ( ph -> ps )

Proof

Step	Hyp	Ref	Expression
1		nfrals.1	|- F/_ x A
2		nfrals.2	|- F/ x ph
3		nfrals.3	|- F/ x ps
4		df-rals	|- ( AE y e. A ( ph -> ps ) <-> ( A. y e. A ( ph -> ps ) /\ E. y e. A ph ) )
5	2 3	nfim	|- F/ x ( ph -> ps )
6	1 5	nfralw	|- F/ x A. y e. A ( ph -> ps )
7	1 2	nfrexw	|- F/ x E. y e. A ph
8	6 7	nfan	|- F/ x ( A. y e. A ( ph -> ps ) /\ E. y e. A ph )
9	4 8	nfxfr	|- F/ x AE y e. A ( ph -> ps )