Consider the following premises. forall x_1: P_1(x_1). exists x_1: Q(x_1). forall x_1: P_1(x_1) -> P_2(x_1). Can we infer exists x_1: P_2(x_1) and 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_2(x_1), forall x_1: P_1(x_1) we can infer forall x_1: P_2(x_1) via universal modus ponens. Finally, from forall x_1: P_2(x_1), exists x_1: Q(x_1) we can infer exists x_1: P_2(x_1) and Q(x_1) via existential conjunction. Therefore, the answer is yes.