Consider the following premises. forall x_3: P_3(x_3) <-> Q_3(x_3). forall x_3: P_4(x_3) -> P_3(x_3). Can we infer exists x_3: ~(P_4(x_3) -> Q_3(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) we can infer forall x_3: P_3(x_3) -> Q_3(x_3) via universal biconditional elimination. Finally, from forall x_3: P_4(x_3) -> P_3(x_3), forall x_3: P_3(x_3) -> Q_3(x_3) we can infer forall x_3: P_4(x_3) -> Q_3(x_3) via universal transitivity, which contradicts exists x_3: ~(P_4(x_3) -> Q_3(x_3)). Therefore, the answer is no.