(** We define a variant of the former existential quantification in order to express the existence of an object that fulfills the conjunction of two predicates P and Q.  *)
Definition ex2 (X : Typ1) (P Q : X -> Prop) : Prop :=
forall E : Prop, (forall x : X, P x -> Q x -> E) -> E.

Lemma ex2_intro :
forall (X : Typ1) (P Q : X -> Prop) (x : X), P x -> Q x -> ex2 X P Q.