2ralbida

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 24-Feb-2004)

	Ref	Expression
Hypotheses	2ralbida.1	$⊢ Ⅎ x φ$
	2ralbida.2	$⊢ Ⅎ y φ$
	2ralbida.3	$⊢ (φ \land (x \in A \land y \in B)) \to (ψ \leftrightarrow χ)$
Assertion	2ralbida	$⊢ φ \to (\forall x \in A \forall y \in B ψ \leftrightarrow \forall x \in A \forall y \in B χ)$

Step	Hyp	Ref	Expression
1		2ralbida.1	$⊢ Ⅎ x φ$
2		2ralbida.2	$⊢ Ⅎ y φ$
3		2ralbida.3	$⊢ (φ \land (x \in A \land y \in B)) \to (ψ \leftrightarrow χ)$
4		nfv	$⊢ Ⅎ y x \in A$
5	2 4	nfan	$⊢ Ⅎ y (φ \land x \in A)$
6	3	anassrs	$⊢ ((φ \land x \in A) \land y \in B) \to (ψ \leftrightarrow χ)$
7	5 6	ralbida	$⊢ (φ \land x \in A) \to (\forall y \in B ψ \leftrightarrow \forall y \in B χ)$
8	1 7	ralbida	$⊢ φ \to (\forall x \in A \forall y \in B ψ \leftrightarrow \forall x \in A \forall y \in B χ)$

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