Consider the following premises. exists x_12: U_3(x_12) -> S(x_12). forall x_12: S(x_12) -> U_3(x_12). forall x_12: ~U_3(x_12). forall x_12: U_3(x_12) -> S(x_12). Can we infer forall x_12: S(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: S(x_12) -> U_3(x_12), forall x_12: U_3(x_12) -> S(x_12) we can infer forall x_12: S(x_12) <-> U_3(x_12) via universal biconditional introduction. From forall x_12: S(x_12) <-> U_3(x_12) we can infer forall x_12: S(x_12) -> U_3(x_12) via universal biconditional elimination. From exists x_12: U_3(x_12) -> S(x_12), forall x_12: S(x_12) -> U_3(x_12) we can infer exists x_12: U_3(x_12) <-> S(x_12) via existential biconditional introduction. From exists x_12: U_3(x_12) <-> S(x_12) we can infer exists x_12: S(x_12) -> U_3(x_12) via existential biconditional elimination. Finally, from exists x_12: S(x_12) -> U_3(x_12), forall x_12: ~U_3(x_12) we can infer exists x_12: ~S(x_12) via existential modus tollens, which contradicts forall x_12: S(x_12). Therefore, the answer is no.