Consider the following premises. forall x_3: P_4(x_3) -> ~Q_3(x_3). forall x_3: P_4(x_3) -> P(x_3). forall x_3: P(x_3) -> P_4(x_3). forall x_3: Q_3(x_3). Can we infer forall x_3: ~P(x_3) from them? If possible, name the inference rules being used at each step.
 From forall x_3: P_4(x_3) -> ~Q_3(x_3), forall x_3: Q_3(x_3) we can infer forall x_3: ~P_4(x_3) via universal modus tollens. From forall x_3: P_4(x_3) -> P(x_3), forall x_3: P(x_3) -> P_4(x_3) we can infer forall x_3: P_4(x_3) <-> P(x_3) via universal biconditional introduction. Finally, from forall x_3: P_4(x_3) <-> P(x_3), forall x_3: ~P_4(x_3) we can infer forall x_3: ~P(x_3) via universal biconditional elimination. Therefore, the answer is yes.