Consider the following premises. exists x_9: W(x_9). forall x_9: R(x_9). forall x_9: T_1(x_9) -> T_2(x_9). forall x_9: S_3(x_9) -> T_1(x_9). forall x_9: R(x_9) -> T_1(x_9). Can we infer forall x_9: ~(T_2(x_9) and W(x_9)) from them? If possible, name the inference rules being used at each step.
 From forall x_9: R(x_9) we can infer forall x_9: R(x_9) or S_3(x_9) via universal addition. From forall x_9: R(x_9) -> T_1(x_9), forall x_9: S_3(x_9) -> T_1(x_9), forall x_9: R(x_9) or S_3(x_9) we can infer forall x_9: T_1(x_9) via universal disjunction elimination. From forall x_9: T_1(x_9) -> T_2(x_9), forall x_9: T_1(x_9) we can infer forall x_9: T_2(x_9) via universal modus ponens. Finally, from forall x_9: T_2(x_9), exists x_9: W(x_9) we can infer exists x_9: T_2(x_9) and W(x_9) via existential conjunction, which contradicts forall x_9: ~(T_2(x_9) and W(x_9)). Therefore, the answer is no.