Consider the following premises. forall x_10: P(x_10) -> T_2(x_10). forall x_10: P(x_10). forall x_10: P_10(x_10) -> Q_10(x_10). forall x_10: P_10(x_10). forall x_10: Q_10(x_10) -> S_3(x_10). Can we infer exists x_10: ~S_3(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_10(x_10) -> Q_10(x_10), forall x_10: P_10(x_10) we can infer forall x_10: Q_10(x_10) via universal modus ponens. From forall x_10: Q_10(x_10) -> S_3(x_10), forall x_10: Q_10(x_10) we can infer forall x_10: S_3(x_10) via universal modus ponens. From forall x_10: P(x_10) -> T_2(x_10), forall x_10: P(x_10) we can infer forall x_10: T_2(x_10) via universal modus ponens. From forall x_10: T_2(x_10), forall x_10: S_3(x_10) we can infer forall x_10: T_2(x_10) and S_3(x_10) via universal conjunction. Finally, from forall x_10: T_2(x_10) and S_3(x_10) we can infer forall x_10: T_2(x_10) via universal simplification, which contradicts exists x_10: ~S_3(x_10). Therefore, the answer is no.