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Metamath Proof Explorer

Theorem ralrimivvva

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014)

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

Proof

Step	Hyp	Ref	Expression
1		ralrimivvva.1	$⊢ (φ \land (x \in A \land y \in B \land z \in C)) \to ψ$
2	1	3anassrs	$⊢ (((φ \land x \in A) \land y \in B) \land z \in C) \to ψ$
3	2	ralrimiva	$⊢ ((φ \land x \in A) \land y \in B) \to \forall z \in C ψ$
4	3	ralrimiva	$⊢ (φ \land x \in A) \to \forall y \in B \forall z \in C ψ$
5	4	ralrimiva	$⊢ φ \to \forall x \in A \forall y \in B \forall z \in C ψ$