Consider the following premises. exists x_16: P_5(x_16) -> P_16(x_16). forall x_16: U(x_16) -> Q_1(x_16). forall x_16: P_16(x_16) <-> Q_16(x_16). forall x_16: ~Q_16(x_16). forall x_16: Q_1(x_16) -> P_5(x_16). Can we infer exists x_16: ~U(x_16) from them? If possible, name the inference rules being used at each step.
 From forall x_16: P_16(x_16) <-> Q_16(x_16), forall x_16: ~Q_16(x_16) we can infer forall x_16: ~P_16(x_16) via universal biconditional elimination. From exists x_16: P_5(x_16) -> P_16(x_16), forall x_16: ~P_16(x_16) we can infer exists x_16: ~P_5(x_16) via existential modus tollens. From forall x_16: U(x_16) -> Q_1(x_16), forall x_16: Q_1(x_16) -> P_5(x_16) we can infer forall x_16: U(x_16) -> P_5(x_16) via universal transitivity. Finally, from forall x_16: U(x_16) -> P_5(x_16), exists x_16: ~P_5(x_16) we can infer exists x_16: ~U(x_16) via existential modus tollens. Therefore, the answer is yes.