Consider the following premises. forall x_2: Q(x_2) -> P_2(x_2). forall x_2: Q_2(x_2) -> Q(x_2). forall x_2: P_2(x_2) -> Q_2(x_2). Can we infer exists x_2: ~(Q(x_2) -> P_2(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), forall x_2: Q_2(x_2) -> Q(x_2) we can infer forall x_2: P_2(x_2) -> Q(x_2) via universal transitivity. From forall x_2: Q(x_2) -> P_2(x_2), forall x_2: P_2(x_2) -> Q(x_2) we can infer forall x_2: Q(x_2) <-> P_2(x_2) via universal biconditional introduction. Finally, from forall x_2: Q(x_2) <-> P_2(x_2) we can infer forall x_2: Q(x_2) -> P_2(x_2) via universal biconditional elimination, which contradicts exists x_2: ~(Q(x_2) -> P_2(x_2)). Therefore, the answer is no.