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	|- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch )
	r19.29vva.2	|- ( ph -> E. x e. A E. y e. B ps )
Assertion	r19.29vva	|- ( ph -> ch )

Proof

Step	Hyp	Ref	Expression
1		r19.29vva.1	|- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch )
2		r19.29vva.2	|- ( ph -> E. x e. A E. y e. B ps )
3	1 2	reximddv2	|- ( ph -> E. x e. A E. y e. B ch )
4		id	|- ( ch -> ch )
5	4	rexlimivw	|- ( E. y e. B ch -> ch )
6	5	reximi	|- ( E. x e. A E. y e. B ch -> E. x e. A ch )
7	4	rexlimivw	|- ( E. x e. A ch -> ch )
8	3 6 7	3syl	|- ( ph -> ch )