Consider the following premises. forall x_2: T_1(x_2) or ~S(x_2). forall x_2: Q_3(x_2) or S(x_2). forall x_2: ~Q_3(x_2). Can we infer exists x_2: ~T_1(x_2) from them? If possible, name the inference rules being used at each step.
 From forall x_2: T_1(x_2) or ~S(x_2) we can infer forall x_2: S(x_2) -> T_1(x_2) via universal rewriting. From forall x_2: Q_3(x_2) or S(x_2), forall x_2: ~Q_3(x_2) we can infer forall x_2: S(x_2) via universal disjunctive syllogism. Finally, from forall x_2: S(x_2) -> T_1(x_2), forall x_2: S(x_2) we can infer forall x_2: T_1(x_2) via universal modus ponens, which contradicts exists x_2: ~T_1(x_2). Therefore, the answer is no.