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, 4-Nov-2024)

	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		idd	|- ( ( x e. A /\ y e. B ) -> ( ch -> ch ) )
5	4	rexlimivv	|- ( E. x e. A E. y e. B ch -> ch )
6	3 5	syl	|- ( ph -> ch )