rexlimdv3d

Description: An extended version of rexlimdvv to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024)

		Ref	Expression
	Hypothesis	rexlimdv3d.1	$⊢ φ \to ((x \in A \land y \in B \land z \in C) \to (ψ \to χ))$
	Assertion	rexlimdv3d	$⊢ φ \to (\exists x \in A \exists y \in B \exists z \in C ψ \to χ)$

Step	Hyp	Ref	Expression
1		rexlimdv3d.1	$⊢ φ \to ((x \in A \land y \in B \land z \in C) \to (ψ \to χ))$
2	1	3expd	$⊢ φ \to (x \in A \to (y \in B \to (z \in C \to (ψ \to χ))))$
3	2	imp4d	$⊢ φ \to ((x \in A \land (y \in B \land z \in C)) \to (ψ \to χ))$
4	3	expdimp	$⊢ (φ \land x \in A) \to ((y \in B \land z \in C) \to (ψ \to χ))$
5	4	rexlimdvv	$⊢ (φ \land x \in A) \to (\exists y \in B \exists z \in C ψ \to χ)$
6	5	rexlimdva	$⊢ φ \to (\exists x \in A \exists y \in B \exists z \in C ψ \to χ)$

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