Consider the following premises. exists x_3: ~Q(x_3). forall x_3: Q_4(x_3) <-> Q(x_3). Can we infer exists x_3: ~Q_4(x_3) from them? If possible, name the inference rules being used at each step.
 From forall x_3: Q_4(x_3) <-> Q(x_3) we can infer forall x_3: Q_4(x_3) -> Q(x_3) via universal biconditional elimination. Finally, from forall x_3: Q_4(x_3) -> Q(x_3), exists x_3: ~Q(x_3) we can infer exists x_3: ~Q_4(x_3) via existential modus tollens. Therefore, the answer is yes.