Consider the following premises. forall x_13: Q_7(x_13) -> P(x_13). forall x_13: P_7(x_13) -> Q_7(x_13). forall x_13: P_13(x_13). forall x_13: P(x_13) -> Q(x_13). forall x_13: P_13(x_13) <-> P(x_13). Can we infer exists x_13: ~Q(x_13) from them? If possible, name the inference rules being used at each step.
 From forall x_13: P_13(x_13) <-> P(x_13) we can infer forall x_13: P_13(x_13) -> P(x_13) via universal biconditional elimination. From forall x_13: P_7(x_13) -> Q_7(x_13), forall x_13: Q_7(x_13) -> P(x_13) we can infer forall x_13: P_7(x_13) -> P(x_13) via universal transitivity. From forall x_13: P_13(x_13) we can infer forall x_13: P_13(x_13) or P_7(x_13) via universal addition. From forall x_13: P_13(x_13) -> P(x_13), forall x_13: P_7(x_13) -> P(x_13), forall x_13: P_13(x_13) or P_7(x_13) we can infer forall x_13: P(x_13) via universal disjunction elimination. Finally, from forall x_13: P(x_13) -> Q(x_13), forall x_13: P(x_13) we can infer forall x_13: Q(x_13) via universal modus ponens, which contradicts exists x_13: ~Q(x_13). Therefore, the answer is no.