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Metamath Proof Explorer

Theorem r2al

Description: Double restricted universal quantification. (Contributed by NM, 19-Nov-1995) Reduce dependencies on axioms. (Revised by Wolf Lammen, 9-Jan-2020)

		Ref	Expression
	Assertion	r2al	$⊢ \forall x \in A \forall y \in B φ \leftrightarrow \forall x \forall y ((x \in A \land y \in B) \to φ)$

Proof

Step	Hyp	Ref	Expression
1		19.21v	$⊢ \forall y (x \in A \to (y \in B \to φ)) \leftrightarrow (x \in A \to \forall y (y \in B \to φ))$
2	1	r2allem	$⊢ \forall x \in A \forall y \in B φ \leftrightarrow \forall x \forall y ((x \in A \land y \in B) \to φ)$