Theorem r19.29vva

Description: A commonly used pattern based on r19.29 , version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017) (Proof shortened by Wolf Lammen, 29-Jun-2023)

	Ref	Expression
Hypotheses	r19.29vva.1	$$⊢\left(\left(\left(\phi \wedge x\in A\right)\wedge y\in B\right)\wedge \psi \right)\to \chi$$
	r19.29vva.2	$$⊢\phi \to \exists x\in A\exists y\in B\psi$$
Assertion	r19.29vva	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		r19.29vva.1	$$⊢\left(\left(\left(\phi \wedge x\in A\right)\wedge y\in B\right)\wedge \psi \right)\to \chi$$
2		r19.29vva.2	$$⊢\phi \to \exists x\in A\exists y\in B\psi$$
3	1 2	reximddv2	$$⊢\phi \to \exists x\in A\exists y\in B\chi$$
4		id	$$⊢\chi \to \chi$$
5	4	rexlimivw	$$⊢\exists y\in B\chi \to \chi$$
6	5	reximi	$$⊢\exists x\in A\exists y\in B\chi \to \exists x\in A\chi$$
7	4	rexlimivw	$$⊢\exists x\in A\chi \to \chi$$
8	3 6 7	3syl	$$⊢\phi \to \chi$$