Consider the following premises. exists x_4: P(x_4) -> U_4(x_4). forall x_4: U_4(x_4) -> P(x_4). forall x_4: Q_4(x_4). forall x_4: P(x_4) -> ~Q_4(x_4). Can we infer exists x_4: ~U_4(x_4) from them? If possible, name the inference rules being used at each step.
 From forall x_4: P(x_4) -> ~Q_4(x_4), forall x_4: Q_4(x_4) we can infer forall x_4: ~P(x_4) via universal modus tollens. From exists x_4: P(x_4) -> U_4(x_4), forall x_4: U_4(x_4) -> P(x_4) we can infer exists x_4: P(x_4) <-> U_4(x_4) via existential biconditional introduction. Finally, from exists x_4: P(x_4) <-> U_4(x_4), forall x_4: ~P(x_4) we can infer exists x_4: ~U_4(x_4) via existential biconditional elimination. Therefore, the answer is yes.