Consider the following premises. forall x_12: P_3(x_12) -> ~P_4(x_12). forall x_12: P_12(x_12) -> P_4(x_12). forall x_12: P_3(x_12) or ~T_3(x_12). forall x_12: Q_12(x_12) -> P_4(x_12). forall x_12: P_12(x_12) or Q_12(x_12). Can we infer forall x_12: ~T_3(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) -> P_4(x_12), forall x_12: Q_12(x_12) -> P_4(x_12), forall x_12: P_12(x_12) or Q_12(x_12) we can infer forall x_12: P_4(x_12) via universal disjunction elimination. From forall x_12: P_3(x_12) -> ~P_4(x_12), forall x_12: P_4(x_12) we can infer forall x_12: ~P_3(x_12) via universal modus tollens. From forall x_12: P_3(x_12) or ~T_3(x_12) we can infer forall x_12: T_3(x_12) -> P_3(x_12) via universal rewriting. Finally, from forall x_12: T_3(x_12) -> P_3(x_12), forall x_12: ~P_3(x_12) we can infer forall x_12: ~T_3(x_12) via universal modus tollens. Therefore, the answer is yes.