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	$$⊢\left(x\in A\wedge y\in B\right)\to \phi$$
Assertion	rgen2	$$⊢\forall x\in A\forall y\in B\phi$$

Proof

Step	Hyp	Ref	Expression
1		rgen2.1	$$⊢\left(x\in A\wedge y\in B\right)\to \phi$$
2	1	ralrimiva	$$⊢x\in A\to \forall y\in B\phi$$
3	2	rgen	$$⊢\forall x\in A\forall y\in B\phi$$