Consider the following premises. forall x: U(x) -> P_3(x). exists x: P_3(x) -> ~P(x). forall x: P_3(x) -> U(x). forall x: P(x). Can we infer exists x: ~U(x) from them? If possible, name the inference rules being used at each step.
 From forall x: P_3(x) -> U(x), forall x: U(x) -> P_3(x) we can infer forall x: P_3(x) <-> U(x) via universal biconditional introduction. From exists x: P_3(x) -> ~P(x), forall x: P(x) we can infer exists x: ~P_3(x) via existential modus tollens. Finally, from forall x: P_3(x) <-> U(x), exists x: ~P_3(x) we can infer exists x: ~U(x) via existential biconditional elimination. Therefore, the answer is yes.