Consider the following premises. forall y: P_1(y) or ~T(y). forall y: P_1(y) -> P(y). forall y: ~P(y). Can we infer exists y: T(y) from them?
 From forall y: P_1(y) -> P(y), forall y: ~P(y) we can infer forall y: ~P_1(y). From forall y: P_1(y) or ~T(y) we can infer forall y: T(y) -> P_1(y). Finally, from forall y: T(y) -> P_1(y), forall y: ~P_1(y) we can infer forall y: ~T(y), which contradicts exists y: T(y). Therefore, the answer is no.