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Metamath Proof Explorer

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	$⊢ Ⅎ y \forall x \in A \forall y \in B φ$

Proof

Step	Hyp	Ref	Expression
1		r2al	$⊢ \forall x \in A \forall y \in B φ \leftrightarrow \forall x \forall y ((x \in A \land y \in B) \to φ)$
2		nfa2	$⊢ Ⅎ y \forall x \forall y ((x \in A \land y \in B) \to φ)$
3	1 2	nfxfr	$⊢ Ⅎ y \forall x \in A \forall y \in B φ$