Consider the following premises. forall x: S_3(x) -> T_2(x). forall x: T(x) <-> T_2(x). exists x: ~T(x). Can we infer exists x: ~S_3(x) from them? If possible, name the inference rules being used at each step.
 From forall x: T(x) <-> T_2(x) we can infer forall x: T_2(x) -> T(x) via universal biconditional elimination. From forall x: S_3(x) -> T_2(x), forall x: T_2(x) -> T(x) we can infer forall x: S_3(x) -> T(x) via universal transitivity. Finally, from forall x: S_3(x) -> T(x), exists x: ~T(x) we can infer exists x: ~S_3(x) via existential modus tollens. Therefore, the answer is yes.