Consider the following premises. forall x: T(x) or S(x). forall x: T(x) <-> U_2(x). exists x: ~U_2(x). Can we infer forall x: ~S(x) from them? If possible, name the inference rules being used at each step.
 From forall x: T(x) <-> U_2(x) we can infer forall x: T(x) -> U_2(x) via universal biconditional elimination. From forall x: T(x) -> U_2(x), exists x: ~U_2(x) we can infer exists x: ~T(x) via existential modus tollens. Finally, from forall x: T(x) or S(x), exists x: ~T(x) we can infer exists x: S(x) via existential disjunctive syllogism, which contradicts forall x: ~S(x). Therefore, the answer is no.