Consider the following premises. exists y: T(y) -> P(y). forall y: ~T_2(y). forall y: R(y) <-> T_2(y). forall y: R(y) <-> T(y). forall y: P(y) -> T(y). Can we infer exists y: ~P(y) from them?
 From forall y: R(y) <-> T_2(y), forall y: ~T_2(y) we can infer forall y: ~R(y). From forall y: P(y) -> T(y), exists y: T(y) -> P(y) we can infer exists y: P(y) <-> T(y). From forall y: R(y) <-> T(y), forall y: ~R(y) we can infer forall y: ~T(y). Finally, from exists y: P(y) <-> T(y), forall y: ~T(y) we can infer exists y: ~P(y). Therefore, the answer is yes.