Consider the following premises. forall x_13: P_4(x_13) or Q_13(x_13). forall x_13: P_1(x_13) <-> Q_13(x_13). forall x_13: P_4(x_13) -> P_1(x_13). forall x_13: P_1(x_13) -> Q_13(x_13). exists x_13: Q(x_13). Can we infer forall x_13: ~(P_1(x_13) and Q(x_13)) from them? If possible, name the inference rules being used at each step.
 From forall x_13: P_1(x_13) <-> Q_13(x_13) we can infer forall x_13: Q_13(x_13) -> P_1(x_13) via universal biconditional elimination. From forall x_13: P_1(x_13) -> Q_13(x_13), forall x_13: Q_13(x_13) -> P_1(x_13) we can infer forall x_13: P_1(x_13) <-> Q_13(x_13) via universal biconditional introduction. From forall x_13: P_1(x_13) <-> Q_13(x_13) we can infer forall x_13: Q_13(x_13) -> P_1(x_13) via universal biconditional elimination. From forall x_13: P_4(x_13) -> P_1(x_13), forall x_13: Q_13(x_13) -> P_1(x_13), forall x_13: P_4(x_13) or Q_13(x_13) we can infer forall x_13: P_1(x_13) via universal disjunction elimination. Finally, from forall x_13: P_1(x_13), exists x_13: Q(x_13) we can infer exists x_13: P_1(x_13) and Q(x_13) via existential conjunction, which contradicts forall x_13: ~(P_1(x_13) and Q(x_13)). Therefore, the answer is no.