Theorem rgen3

Description: Generalization rule for restricted quantification, with three quantifiers. (Contributed by NM, 12-Jan-2008)

	Ref	Expression
Hypothesis	rgen3.1	$$⊢\left(x\in A\wedge y\in B\wedge z\in C\right)\to \phi$$
Assertion	rgen3	$$⊢\forall x\in A\forall y\in B\forall z\in C\phi$$

Proof

Step	Hyp	Ref	Expression
1		rgen3.1	$$⊢\left(x\in A\wedge y\in B\wedge z\in C\right)\to \phi$$
2	1	3expa	$$⊢\left(\left(x\in A\wedge y\in B\right)\wedge z\in C\right)\to \phi$$
3	2	ralrimiva	$$⊢\left(x\in A\wedge y\in B\right)\to \forall z\in C\phi$$
4	3	rgen2	$$⊢\forall x\in A\forall y\in B\forall z\in C\phi$$