Consider the following premises. exists x: ~U(x). forall x: T_4(x) -> P_3(x). forall x: P_3(x) -> U(x). forall x: U(x) -> T_4(x). Can we infer exists x: ~P_3(x) from them?
 From forall x: U(x) -> T_4(x), forall x: T_4(x) -> P_3(x) we can infer forall x: U(x) -> P_3(x). 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). Finally, from forall x: P_3(x) <-> U(x), exists x: ~U(x) we can infer exists x: ~P_3(x). Therefore, the answer is yes.