Consider the following premises. exists x_12: Q_12(x_12) -> R_12(x_12). forall x_12: P_12(x_12) -> R_12(x_12). forall x_12: P_1(x_12). forall x_12: P_1(x_12) -> T_3(x_12). forall x_12: P_12(x_12) or Q_12(x_12). Can we infer exists x_12: T_3(x_12) and R_12(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) -> R_12(x_12), exists x_12: Q_12(x_12) -> R_12(x_12), forall x_12: P_12(x_12) or Q_12(x_12) we can infer exists x_12: R_12(x_12) via existential disjunction elimination. From forall x_12: P_1(x_12) -> T_3(x_12), forall x_12: P_1(x_12) we can infer forall x_12: T_3(x_12) via universal modus ponens. Finally, from forall x_12: T_3(x_12), exists x_12: R_12(x_12) we can infer exists x_12: T_3(x_12) and R_12(x_12) via existential conjunction. Therefore, the answer is yes.