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Metamath Proof Explorer

Theorem ralxp

Description: Universal quantification restricted to a Cartesian product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004) (Revised by Mario Carneiro, 29-Dec-2014)

		Ref	Expression
	Hypothesis	ralxp.1	$⊢ x = ⟨y, z⟩ \to (φ \leftrightarrow ψ)$
	Assertion	ralxp	$⊢ \forall x \in (A \times B) φ \leftrightarrow \forall y \in A \forall z \in B ψ$

Proof

Step	Hyp	Ref	Expression
1		ralxp.1	$⊢ x = ⟨y, z⟩ \to (φ \leftrightarrow ψ)$
2		iunxpconst	$⊢ ⋃_{y \in A} (\{y\} \times B) = A \times B$
3	2	raleqi	$⊢ \forall x \in ⋃_{y \in A} (\{y\} \times B) φ \leftrightarrow \forall x \in (A \times B) φ$
4	1	raliunxp	$⊢ \forall x \in ⋃_{y \in A} (\{y\} \times B) φ \leftrightarrow \forall y \in A \forall z \in B ψ$
5	3 4	bitr3i	$⊢ \forall x \in (A \times B) φ \leftrightarrow \forall y \in A \forall z \in B ψ$