Consider the following premises. forall x_5: P_5(x_5). forall x_5: R_4(x_5) or S(x_5). forall x_5: ~R_4(x_5). forall x_5: P_5(x_5) -> T(x_5). Can we infer exists x_5: ~(T(x_5) and S(x_5)) from them? If possible, name the inference rules being used at each step.
 From forall x_5: P_5(x_5) -> T(x_5), forall x_5: P_5(x_5) we can infer forall x_5: T(x_5) via universal modus ponens. From forall x_5: R_4(x_5) or S(x_5), forall x_5: ~R_4(x_5) we can infer forall x_5: S(x_5) via universal disjunctive syllogism. Finally, from forall x_5: T(x_5), forall x_5: S(x_5) we can infer forall x_5: T(x_5) and S(x_5) via universal conjunction, which contradicts exists x_5: ~(T(x_5) and S(x_5)). Therefore, the answer is no.