Consider the following premises. forall x_13: P_8(x_13) -> Q_13(x_13). forall x_13: S_2(x_13) -> T(x_13). forall x_13: S_2(x_13). forall x_13: T(x_13) or ~Q_13(x_13). exists x_13: P(x_13) -> P_8(x_13). Can we infer exists x_13: T(x_13) from them? If possible, name the inference rules being used at each step.
 From forall x_13: T(x_13) or ~Q_13(x_13) we can infer forall x_13: Q_13(x_13) -> T(x_13) via universal rewriting. From forall x_13: P_8(x_13) -> Q_13(x_13), forall x_13: Q_13(x_13) -> T(x_13) we can infer forall x_13: P_8(x_13) -> T(x_13) via universal transitivity. From forall x_13: S_2(x_13) we can infer forall x_13: S_2(x_13) or P(x_13) via universal addition. From exists x_13: P(x_13) -> P_8(x_13), forall x_13: P_8(x_13) -> T(x_13) we can infer exists x_13: P(x_13) -> T(x_13) via existential transitivity. Finally, from forall x_13: S_2(x_13) -> T(x_13), exists x_13: P(x_13) -> T(x_13), forall x_13: S_2(x_13) or P(x_13) we can infer exists x_13: T(x_13) via existential disjunction elimination. Therefore, the answer is yes.