Consider the following premises. exists x: Q(x). forall x: Q_1(x). forall x: Q_1(x) <-> T(x). Can we infer exists x: Q(x) and T(x) from them?
 From forall x: Q_1(x) <-> T(x) we can infer forall x: Q_1(x) -> T(x). From forall x: Q_1(x) -> T(x), forall x: Q_1(x) we can infer forall x: T(x). Finally, from exists x: Q(x), forall x: T(x) we can infer exists x: Q(x) and T(x). Therefore, the answer is yes.