Consider the following premises. forall x_8: Q_8(x_8) -> T(x_8). forall x_8: ~P_8(x_8). forall x_8: P_8(x_8) or Q_8(x_8). forall x_8: Q_5(x_8) -> T(x_8). Can we infer forall x_8: T(x_8) or S(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: P_8(x_8) or Q_8(x_8), forall x_8: ~P_8(x_8) we can infer forall x_8: Q_8(x_8) via universal disjunctive syllogism. From forall x_8: Q_8(x_8) we can infer forall x_8: Q_8(x_8) or Q_5(x_8) via universal addition. From forall x_8: Q_8(x_8) -> T(x_8), forall x_8: Q_5(x_8) -> T(x_8), forall x_8: Q_8(x_8) or Q_5(x_8) we can infer forall x_8: T(x_8) via universal disjunction elimination. Finally, from forall x_8: T(x_8) we can infer forall x_8: T(x_8) or S(x_8) via universal addition. Therefore, the answer is yes.