Consider the following premises. exists x_8: ~S_2(x_8). forall x_8: Q_2(x_8) -> S_2(x_8). forall x_8: U(x_8) -> Q_2(x_8). forall x_8: S_2(x_8) <-> U(x_8). Can we infer forall x_8: U(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: S_2(x_8) <-> U(x_8) we can infer forall x_8: S_2(x_8) -> U(x_8) via universal biconditional elimination. From forall x_8: U(x_8) -> Q_2(x_8), forall x_8: Q_2(x_8) -> S_2(x_8) we can infer forall x_8: U(x_8) -> S_2(x_8) via universal transitivity. From forall x_8: U(x_8) -> S_2(x_8), forall x_8: S_2(x_8) -> U(x_8) we can infer forall x_8: U(x_8) <-> S_2(x_8) via universal biconditional introduction. Finally, from forall x_8: U(x_8) <-> S_2(x_8), exists x_8: ~S_2(x_8) we can infer exists x_8: ~U(x_8) via existential biconditional elimination, which contradicts forall x_8: U(x_8). Therefore, the answer is no.