Metamath Proof Explorer

Theorem reximddv2

Description: Double deduction from Theorem 19.22 of Margaris p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019)

	Ref	Expression
Hypotheses	reximddv2.1	$⊢ (((φ \land x \in A) \land y \in B) \land ψ) \to χ$
	reximddv2.2	$⊢ φ \to \exists x \in A \exists y \in B ψ$
Assertion	reximddv2	$⊢ φ \to \exists x \in A \exists y \in B χ$

Step	Hyp	Ref	Expression
1		reximddv2.1	$⊢ (((φ \land x \in A) \land y \in B) \land ψ) \to χ$
2		reximddv2.2	$⊢ φ \to \exists x \in A \exists y \in B ψ$
3	1	ex	$⊢ ((φ \land x \in A) \land y \in B) \to (ψ \to χ)$
4	3	reximdva	$⊢ (φ \land x \in A) \to (\exists y \in B ψ \to \exists y \in B χ)$
5	4	impr	$⊢ (φ \land (x \in A \land \exists y \in B ψ)) \to \exists y \in B χ$
6	5 2	reximddv	$⊢ φ \to \exists x \in A \exists y \in B χ$