Consider the following premises. forall x_18: P_4(x_18) or P_18(x_18). forall x_18: T_1(x_18) -> P_18(x_18). forall x_18: P_4(x_18) -> T_1(x_18). forall x_18: P_9(x_18) or Q_9(x_18). forall x_18: P_18(x_18) -> T_1(x_18). exists x_18: ~P_9(x_18). Can we infer exists x_18: T_1(x_18) and Q_9(x_18) from them? If possible, name the inference rules being used at each step.
 From forall x_18: P_18(x_18) -> T_1(x_18), forall x_18: T_1(x_18) -> P_18(x_18) we can infer forall x_18: P_18(x_18) <-> T_1(x_18) via universal biconditional introduction. From forall x_18: P_18(x_18) <-> T_1(x_18) we can infer forall x_18: P_18(x_18) -> T_1(x_18) via universal biconditional elimination. From forall x_18: P_9(x_18) or Q_9(x_18), exists x_18: ~P_9(x_18) we can infer exists x_18: Q_9(x_18) via existential disjunctive syllogism. From forall x_18: P_4(x_18) -> T_1(x_18), forall x_18: P_18(x_18) -> T_1(x_18), forall x_18: P_4(x_18) or P_18(x_18) we can infer forall x_18: T_1(x_18) via universal disjunction elimination. Finally, from forall x_18: T_1(x_18), exists x_18: Q_9(x_18) we can infer exists x_18: T_1(x_18) and Q_9(x_18) via existential conjunction. Therefore, the answer is yes.