Consider the following premises. forall x_13: P_13(x_13) <-> Q_13(x_13). forall x_13: ~P_10(x_13) or U(x_13). exists x_13: S(x_13) <-> Q_13(x_13). forall x_13: P_10(x_13) or S(x_13). forall x_13: ~P_13(x_13). Can we infer exists x_13: U(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: ~P_13(x_13) we can infer forall x_13: ~Q_13(x_13) via universal biconditional elimination. From forall x_13: P_10(x_13) or S(x_13), forall x_13: ~P_10(x_13) or U(x_13) we can infer forall x_13: S(x_13) or U(x_13) via universal resolution. From exists x_13: S(x_13) <-> Q_13(x_13), forall x_13: ~Q_13(x_13) we can infer exists x_13: ~S(x_13) via existential biconditional elimination. Finally, from forall x_13: S(x_13) or U(x_13), exists x_13: ~S(x_13) we can infer exists x_13: U(x_13) via existential disjunctive syllogism. Therefore, the answer is yes.