Consider the following premises. forall x_17: P_1(x_17) -> P(x_17). forall x_17: P_1(x_17). forall x_17: ~P_15(x_17) or R_15(x_17). forall x_17: R_15(x_17) -> T_4(x_17). forall x_17: P_17(x_17) -> Q_17(x_17). exists x_17: Q_17(x_17) -> T_4(x_17). forall x_17: P_15(x_17) or P_17(x_17). Can we infer exists x_17: T_4(x_17) and P(x_17) from them? If possible, name the inference rules being used at each step.
 From forall x_17: P_17(x_17) -> Q_17(x_17), exists x_17: Q_17(x_17) -> T_4(x_17) we can infer exists x_17: P_17(x_17) -> T_4(x_17) via existential transitivity. From forall x_17: P_15(x_17) or P_17(x_17), forall x_17: ~P_15(x_17) or R_15(x_17) we can infer forall x_17: P_17(x_17) or R_15(x_17) via universal resolution. From exists x_17: P_17(x_17) -> T_4(x_17), forall x_17: R_15(x_17) -> T_4(x_17), forall x_17: P_17(x_17) or R_15(x_17) we can infer exists x_17: T_4(x_17) via existential disjunction elimination. From forall x_17: P_1(x_17) -> P(x_17), forall x_17: P_1(x_17) we can infer forall x_17: P(x_17) via universal modus ponens. Finally, from exists x_17: T_4(x_17), forall x_17: P(x_17) we can infer exists x_17: T_4(x_17) and P(x_17) via existential conjunction. Therefore, the answer is yes.