Consider the following premises. exists x_2: ~Q_2(x_2). forall x_2: P_2(x_2) -> Q_2(x_2). forall x_2: P_2(x_2) <-> Q(x_2). Can we infer exists x_2: ~Q(x_2) from them? If possible, name the inference rules being used at each step.
 From forall x_2: P_2(x_2) -> Q_2(x_2), exists x_2: ~Q_2(x_2) we can infer exists x_2: ~P_2(x_2) via existential modus tollens. Finally, from forall x_2: P_2(x_2) <-> Q(x_2), exists x_2: ~P_2(x_2) we can infer exists x_2: ~Q(x_2) via existential biconditional elimination. Therefore, the answer is yes.