Consider the following premises. forall x_13: P(x_13) -> ~Q_2(x_13). forall x_13: P(x_13) -> U_4(x_13). forall x_13: Q_2(x_13). forall x_13: P(x_13) -> U_4(x_13). exists x_13: U_4(x_13) -> P(x_13). Can we infer exists x_13: ~U_4(x_13) from them? If possible, name the inference rules being used at each step.
 From forall x_13: P(x_13) -> U_4(x_13), exists x_13: U_4(x_13) -> P(x_13) we can infer exists x_13: P(x_13) <-> U_4(x_13) via existential biconditional introduction. From exists x_13: P(x_13) <-> U_4(x_13) we can infer exists x_13: U_4(x_13) -> P(x_13) via existential biconditional elimination. From forall x_13: P(x_13) -> U_4(x_13), exists x_13: U_4(x_13) -> P(x_13) we can infer exists x_13: P(x_13) <-> U_4(x_13) via existential biconditional introduction. From forall x_13: P(x_13) -> ~Q_2(x_13), forall x_13: Q_2(x_13) we can infer forall x_13: ~P(x_13) via universal modus tollens. Finally, from exists x_13: P(x_13) <-> U_4(x_13), forall x_13: ~P(x_13) we can infer exists x_13: ~U_4(x_13) via existential biconditional elimination. Therefore, the answer is yes.