Theorem nfra2w

Description: Similar to Lemma 24 of Monk2 p. 114, except that quantification is restricted. Once derived from hbra2VD . Version of nfra2 with a disjoint variable condition not requiring ax-13 . (Contributed by Alan Sare, 31-Dec-2011) Reduce axiom usage. (Revised by GG, 24-Sep-2024) (Proof shortened by Wolf Lammen, 3-Jan-2025)

	Ref	Expression
Assertion	nfra2w	$$⊢Ⅎy\forall x\in A\forall y\in B\phi$$

Proof

Step	Hyp	Ref	Expression
1		r2al	$$⊢\forall x\in A\forall y\in B\phi \leftrightarrow \forall x\forall y\left(\left(x\in A\wedge y\in B\right)\to \phi \right)$$
2		nfa2	$$⊢Ⅎy\forall x\forall y\left(\left(x\in A\wedge y\in B\right)\to \phi \right)$$
3	1 2	nfxfr	$$⊢Ⅎy\forall x\in A\forall y\in B\phi$$