Consider the following premises. exists x: Q_1(x) -> S(x). forall x: T_1(x) -> Q(x). forall x: Q(x) -> Q_1(x). Can we infer exists x: T_1(x) -> S(x) from them? If possible, name the inference rules being used at each step.
 From forall x: Q(x) -> Q_1(x), exists x: Q_1(x) -> S(x) we can infer exists x: Q(x) -> S(x) via existential transitivity. Finally, from forall x: T_1(x) -> Q(x), exists x: Q(x) -> S(x) we can infer exists x: T_1(x) -> S(x) via existential transitivity. Therefore, the answer is yes.