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Metamath Proof Explorer

Theorem rgen3

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

		Ref	Expression
	Hypothesis	rgen3.1	$⊢ (x \in A \land y \in B \land z \in C) \to φ$
	Assertion	rgen3	$⊢ \forall x \in A \forall y \in B \forall z \in C φ$

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