Consider the following premises. exists x: S_4(x) <-> U(x). forall x: S_4(x). forall x: ~U(x). Can we infer exists x: T(x) from them? If possible, name the inference rules being used at each step.
 From exists x: S_4(x) <-> U(x), forall x: ~U(x) we can infer exists x: ~S_4(x) via existential biconditional elimination. From forall x: S_4(x) we can infer forall x: S_4(x) or T(x) via universal addition. Finally, from forall x: S_4(x) or T(x), exists x: ~S_4(x) we can infer exists x: T(x) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.