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