Consider the following premises. forall y: P_4(y) -> T(y). forall y: T(y) -> P(y). exists y: S(y) -> T(y). forall y: P_4(y) or S(y). Can we infer forall y: ~P(y) from them? If possible, name the inference rules being used at each step.
 From forall y: P_4(y) -> T(y), exists y: S(y) -> T(y), forall y: P_4(y) or S(y) we can infer exists y: T(y) via existential disjunction elimination. Finally, from forall y: T(y) -> P(y), exists y: T(y) we can infer exists y: P(y) via existential modus ponens, which contradicts forall y: ~P(y). Therefore, the answer is no.