Metamath Proof Explorer

Theorem r19.29vvaOLD

Description: Obsolete version of r19.29vva as of 28-Jun-2023. (Contributed by Thierry Arnoux, 26-Nov-2017) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	r19.29vva.1	$⊢ (((φ \land x \in A) \land y \in B) \land ψ) \to χ$
		r19.29vva.2	$⊢ φ \to \exists x \in A \exists y \in B ψ$
	Assertion	r19.29vvaOLD	$⊢ φ \to χ$

Proof

Step	Hyp	Ref	Expression
1		r19.29vva.1	$⊢ (((φ \land x \in A) \land y \in B) \land ψ) \to χ$
2		r19.29vva.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	ralrimiva	$⊢ (φ \land x \in A) \to \forall y \in B (ψ \to χ)$
5	4	ralrimiva	$⊢ φ \to \forall x \in A \forall y \in B (ψ \to χ)$
6	5 2	r19.29d2r	$⊢ φ \to \exists x \in A \exists y \in B ((ψ \to χ) \land ψ)$
7		pm3.35	$⊢ (ψ \land (ψ \to χ)) \to χ$
8	7	ancoms	$⊢ ((ψ \to χ) \land ψ) \to χ$
9	8	rexlimivw	$⊢ \exists y \in B ((ψ \to χ) \land ψ) \to χ$
10	9	rexlimivw	$⊢ \exists x \in A \exists y \in B ((ψ \to χ) \land ψ) \to χ$
11	6 10	syl	$⊢ φ \to χ$