Consider the following premises. forall x_3: P_2(x_3) -> P_3(x_3). forall x_3: ~Q_3(x_3). exists x_3: P_3(x_3) -> P_2(x_3). forall x_3: P_3(x_3) -> Q_3(x_3). Can we infer exists x_3: ~P_2(x_3) from them? If possible, name the inference rules being used at each step.
 From forall x_3: P_3(x_3) -> Q_3(x_3), forall x_3: ~Q_3(x_3) we can infer forall x_3: ~P_3(x_3) via universal modus tollens. From forall x_3: P_2(x_3) -> P_3(x_3), exists x_3: P_3(x_3) -> P_2(x_3) we can infer exists x_3: P_2(x_3) <-> P_3(x_3) via existential biconditional introduction. Finally, from exists x_3: P_2(x_3) <-> P_3(x_3), forall x_3: ~P_3(x_3) we can infer exists x_3: ~P_2(x_3) via existential biconditional elimination. Therefore, the answer is yes.