Consider the following premises. forall y: U_3(y). exists y: P_2(y) -> ~T(y). forall y: T(y) <-> U_3(y). Can we infer exists y: ~P_2(y) from them?
 From forall y: T(y) <-> U_3(y) we can infer forall y: U_3(y) -> T(y). From forall y: U_3(y) -> T(y), forall y: U_3(y) we can infer forall y: T(y). Finally, from exists y: P_2(y) -> ~T(y), forall y: T(y) we can infer exists y: ~P_2(y). Therefore, the answer is yes.