Consider the following premises. forall x_9: P_1(x_9) or T_3(x_9). forall x_9: T_3(x_9) -> U_1(x_9). forall x_9: T(x_9). exists x_9: W(x_9) -> ~U_1(x_9). forall x_9: P_1(x_9) -> U_1(x_9). Can we infer forall x_9: W(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_1(x_9) -> U_1(x_9), forall x_9: T_3(x_9) -> U_1(x_9), forall x_9: P_1(x_9) or T_3(x_9) we can infer forall x_9: U_1(x_9) via universal disjunction elimination. From forall x_9: U_1(x_9), forall x_9: T(x_9) we can infer forall x_9: U_1(x_9) and T(x_9) via universal conjunction. From forall x_9: U_1(x_9) and T(x_9) we can infer forall x_9: U_1(x_9) via universal simplification. Finally, from exists x_9: W(x_9) -> ~U_1(x_9), forall x_9: U_1(x_9) we can infer exists x_9: ~W(x_9) via existential modus tollens, which contradicts forall x_9: W(x_9). Therefore, the answer is no.