Consider the following premises. forall x: Q_2(x) -> S_2(x). forall x: T(x) <-> U(x). forall x: ~T(x). forall x: S_2(x) -> Q_2(x). exists x: U(x) <-> Q_2(x). Can we infer forall x: S_2(x) from them? If possible, name the inference rules being used at each step.
 From forall x: T(x) <-> U(x), forall x: ~T(x) we can infer forall x: ~U(x) via universal biconditional elimination. From exists x: U(x) <-> Q_2(x), forall x: ~U(x) we can infer exists x: ~Q_2(x) via existential biconditional elimination. From forall x: S_2(x) -> Q_2(x), forall x: Q_2(x) -> S_2(x) we can infer forall x: S_2(x) <-> Q_2(x) via universal biconditional introduction. Finally, from forall x: S_2(x) <-> Q_2(x), exists x: ~Q_2(x) we can infer exists x: ~S_2(x) via existential biconditional elimination, which contradicts forall x: S_2(x). Therefore, the answer is no.