Theorem rgen3

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

	Ref	Expression
Hypothesis	rgen3.1	|- ( ( x e. A /\ y e. B /\ z e. C ) -> ph )
Assertion	rgen3	|- A. x e. A A. y e. B A. z e. C ph

Proof

Step	Hyp	Ref	Expression
1		rgen3.1	|- ( ( x e. A /\ y e. B /\ z e. C ) -> ph )
2	1	3expa	|- ( ( ( x e. A /\ y e. B ) /\ z e. C ) -> ph )
3	2	ralrimiva	|- ( ( x e. A /\ y e. B ) -> A. z e. C ph )
4	3	rgen2	|- A. x e. A A. y e. B A. z e. C ph