Consider the following premises. forall x_5: R_1(x_5). forall x_5: Q_5(x_5) -> T_2(x_5). forall x_5: P_5(x_5) -> T_2(x_5). forall x_5: P_5(x_5) or Q_5(x_5). Can we infer exists x_5: ~T_2(x_5) from them? If possible, name the inference rules being used at each step.
 From forall x_5: P_5(x_5) -> T_2(x_5), forall x_5: Q_5(x_5) -> T_2(x_5), forall x_5: P_5(x_5) or Q_5(x_5) we can infer forall x_5: T_2(x_5) via universal disjunction elimination. From forall x_5: R_1(x_5), forall x_5: T_2(x_5) we can infer forall x_5: R_1(x_5) and T_2(x_5) via universal conjunction. Finally, from forall x_5: R_1(x_5) and T_2(x_5) we can infer forall x_5: R_1(x_5) via universal simplification, which contradicts exists x_5: ~T_2(x_5). Therefore, the answer is no.