Consider the following premises. forall x_12: P_5(x_12). forall x_12: P_5(x_12) -> Q_12(x_12). forall x_12: W(x_12) <-> Q_12(x_12). Can we infer exists x_12: ~(W(x_12) or Q(x_12)) from them? If possible, name the inference rules being used at each step.
 From forall x_12: W(x_12) <-> Q_12(x_12) we can infer forall x_12: Q_12(x_12) -> W(x_12) via universal biconditional elimination. From forall x_12: P_5(x_12) -> Q_12(x_12), forall x_12: P_5(x_12) we can infer forall x_12: Q_12(x_12) via universal modus ponens. From forall x_12: Q_12(x_12) -> W(x_12), forall x_12: Q_12(x_12) we can infer forall x_12: W(x_12) via universal modus ponens. Finally, from forall x_12: W(x_12) we can infer forall x_12: W(x_12) or Q(x_12) via universal addition, which contradicts exists x_12: ~(W(x_12) or Q(x_12)). Therefore, the answer is no.