Consider the following premises. forall x_1: Q_2(x_1) -> Q_3(x_1). forall x_1: P(x_1). forall x_1: Q_2(x_1). Can we infer exists x_1: ~Q_3(x_1) from them? If possible, name the inference rules being used at each step.
 From forall x_1: Q_2(x_1), forall x_1: P(x_1) we can infer forall x_1: Q_2(x_1) and P(x_1) via universal conjunction. From forall x_1: Q_2(x_1) and P(x_1) we can infer forall x_1: Q_2(x_1) via universal simplification. Finally, from forall x_1: Q_2(x_1) -> Q_3(x_1), forall x_1: Q_2(x_1) we can infer forall x_1: Q_3(x_1) via universal modus ponens, which contradicts exists x_1: ~Q_3(x_1). Therefore, the answer is no.