Consider the following premises. exists x_1: P(x_1). forall x_1: Q_1(x_1) -> P_3(x_1). forall x_1: P(x_1) -> Q_1(x_1). Can we infer forall x_1: ~P_3(x_1) from them? If possible, name the inference rules being used at each step.
 From forall x_1: P(x_1) -> Q_1(x_1), forall x_1: Q_1(x_1) -> P_3(x_1) we can infer forall x_1: P(x_1) -> P_3(x_1) via universal transitivity. Finally, from forall x_1: P(x_1) -> P_3(x_1), exists x_1: P(x_1) we can infer exists x_1: P_3(x_1) via existential modus ponens, which contradicts forall x_1: ~P_3(x_1). Therefore, the answer is no.