Consider the following premises. forall x_12: R_12(x_12) -> Q_5(x_12). exists x_12: T(x_12) -> ~R_5(x_12). forall x_12: Q_12(x_12) -> R_12(x_12). forall x_12: P_12(x_12) or Q_12(x_12). forall x_12: P_12(x_12) -> R_12(x_12). forall x_12: Q_5(x_12) -> R_5(x_12). Can we infer forall x_12: T(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) -> R_12(x_12), forall x_12: Q_12(x_12) -> R_12(x_12), forall x_12: P_12(x_12) or Q_12(x_12) we can infer forall x_12: R_12(x_12) via universal disjunction elimination. From forall x_12: R_12(x_12) -> Q_5(x_12), forall x_12: Q_5(x_12) -> R_5(x_12) we can infer forall x_12: R_12(x_12) -> R_5(x_12) via universal transitivity. From forall x_12: R_12(x_12) -> R_5(x_12), forall x_12: R_12(x_12) we can infer forall x_12: R_5(x_12) via universal modus ponens. Finally, from exists x_12: T(x_12) -> ~R_5(x_12), forall x_12: R_5(x_12) we can infer exists x_12: ~T(x_12) via existential modus tollens, which contradicts forall x_12: T(x_12). Therefore, the answer is no.