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Metamath Proof Explorer

Theorem reximdvva

Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.22 of Margaris p. 90. (Contributed by AV, 5-Jan-2022)

		Ref	Expression
	Hypothesis	ralimdvva.1	$⊢ (φ \land (x \in A \land y \in B)) \to (ψ \to χ)$
	Assertion	reximdvva	$⊢ φ \to (\exists x \in A \exists y \in B ψ \to \exists x \in A \exists y \in B χ)$

Proof

Step	Hyp	Ref	Expression
1		ralimdvva.1	$⊢ (φ \land (x \in A \land y \in B)) \to (ψ \to χ)$
2	1	anassrs	$⊢ ((φ \land x \in A) \land y \in B) \to (ψ \to χ)$
3	2	reximdva	$⊢ (φ \land x \in A) \to (\exists y \in B ψ \to \exists y \in B χ)$
4	3	reximdva	$⊢ φ \to (\exists x \in A \exists y \in B ψ \to \exists x \in A \exists y \in B χ)$