Consider the following premises. forall x_13: Q_13(x_13) -> P_13(x_13). forall x_13: P_13(x_13) -> Q_13(x_13). forall x_13: P_5(x_13) -> Q_13(x_13). forall x_13: W_3(x_13) <-> P_5(x_13). exists x_13: U_3(x_13) <-> W_3(x_13). forall x_13: ~P_13(x_13). Can we infer forall x_13: U_3(x_13) from them? If possible, name the inference rules being used at each step.
 From forall x_13: P_13(x_13) -> Q_13(x_13), forall x_13: Q_13(x_13) -> P_13(x_13) we can infer forall x_13: P_13(x_13) <-> Q_13(x_13) via universal biconditional introduction. From forall x_13: P_13(x_13) <-> Q_13(x_13), forall x_13: ~P_13(x_13) we can infer forall x_13: ~Q_13(x_13) via universal biconditional elimination. From forall x_13: P_5(x_13) -> Q_13(x_13), forall x_13: ~Q_13(x_13) we can infer forall x_13: ~P_5(x_13) via universal modus tollens. From forall x_13: W_3(x_13) <-> P_5(x_13), forall x_13: ~P_5(x_13) we can infer forall x_13: ~W_3(x_13) via universal biconditional elimination. Finally, from exists x_13: U_3(x_13) <-> W_3(x_13), forall x_13: ~W_3(x_13) we can infer exists x_13: ~U_3(x_13) via existential biconditional elimination, which contradicts forall x_13: U_3(x_13). Therefore, the answer is no.