Consider the following premises. exists x_10: P(x_10) -> Q_3(x_10). forall x_10: Q_3(x_10) -> U_2(x_10). forall x_10: U_2(x_10) -> P(x_10). forall x_10: P_10(x_10) <-> U_2(x_10). forall x_10: ~P_10(x_10). Can we infer exists x_10: ~P(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_10(x_10) <-> U_2(x_10), forall x_10: ~P_10(x_10) we can infer forall x_10: ~U_2(x_10) via universal biconditional elimination. From exists x_10: P(x_10) -> Q_3(x_10), forall x_10: Q_3(x_10) -> U_2(x_10) we can infer exists x_10: P(x_10) -> U_2(x_10) via existential transitivity. From forall x_10: U_2(x_10) -> P(x_10), exists x_10: P(x_10) -> U_2(x_10) we can infer exists x_10: U_2(x_10) <-> P(x_10) via existential biconditional introduction. Finally, from exists x_10: U_2(x_10) <-> P(x_10), forall x_10: ~U_2(x_10) we can infer exists x_10: ~P(x_10) via existential biconditional elimination. Therefore, the answer is yes.