Consider the following premises. forall x_1: P(x_1) -> Q(x_1). forall x_1: P_1(x_1). forall x_1: P_1(x_1) -> P(x_1). Can we infer exists x_1: ~Q(x_1) from them? If possible, name the inference rules being used at each step.
 From forall x_1: P_1(x_1) -> P(x_1), forall x_1: P_1(x_1) we can infer forall x_1: P(x_1) via universal modus ponens. Finally, from forall x_1: P(x_1) -> Q(x_1), forall x_1: P(x_1) we can infer forall x_1: Q(x_1) via universal modus ponens, which contradicts exists x_1: ~Q(x_1). Therefore, the answer is no.