Theorem nfra2w

Description: Similar to Lemma 24 of Monk2 p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD . Version of nfra2 with a disjoint variable condition not requiring ax-13 . (Contributed by Alan Sare, 31-Dec-2011) (Revised by Gino Giotto, 24-Sep-2024) (Proof shortened by Wolf Lammen, 31-Oct-2024)

	Ref	Expression
Assertion	nfra2w	|- F/ y A. x e. A A. y e. B ph

Proof

Step	Hyp	Ref	Expression
1		r2al	|- ( A. x e. A A. y e. B ph <-> A. x A. y ( ( x e. A /\ y e. B ) -> ph ) )
2		alcom	|- ( A. x A. y ( ( x e. A /\ y e. B ) -> ph ) <-> A. y A. x ( ( x e. A /\ y e. B ) -> ph ) )
3	1 2	bitri	|- ( A. x e. A A. y e. B ph <-> A. y A. x ( ( x e. A /\ y e. B ) -> ph ) )
4		nfa1	|- F/ y A. y A. x ( ( x e. A /\ y e. B ) -> ph )
5	3 4	nfxfr	|- F/ y A. x e. A A. y e. B ph