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Metamath Proof Explorer

Theorem rgen2

Description: Generalization rule for restricted quantification, with two quantifiers. This theorem should be used in place of rgen2a since it depends on a smaller set of axioms. (Contributed by NM, 30-May-1999)

		Ref	Expression
	Hypothesis	rgen2.1	$⊢ (x \in A \land y \in B) \to φ$
	Assertion	rgen2	$⊢ \forall x \in A \forall y \in B φ$

Proof

Step	Hyp	Ref	Expression
1		rgen2.1	$⊢ (x \in A \land y \in B) \to φ$
2	1	ralrimiva	$⊢ x \in A \to \forall y \in B φ$
3	2	rgen	$⊢ \forall x \in A \forall y \in B φ$