Consider the following premises. forall x_12: P_12(x_12) <-> Q_12(x_12). forall x_12: ~P_12(x_12). forall x_12: Q_12(x_12) or Q_7(x_12). forall x_12: Q_7(x_12) -> S_2(x_12). forall x_12: U_1(x_12). Can we infer exists x_12: ~U_1(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) <-> Q_12(x_12), forall x_12: ~P_12(x_12) we can infer forall x_12: ~Q_12(x_12) via universal biconditional elimination. From forall x_12: Q_12(x_12) or Q_7(x_12), forall x_12: ~Q_12(x_12) we can infer forall x_12: Q_7(x_12) via universal disjunctive syllogism. From forall x_12: Q_7(x_12) -> S_2(x_12), forall x_12: Q_7(x_12) we can infer forall x_12: S_2(x_12) via universal modus ponens. From forall x_12: U_1(x_12), forall x_12: S_2(x_12) we can infer forall x_12: U_1(x_12) and S_2(x_12) via universal conjunction. Finally, from forall x_12: U_1(x_12) and S_2(x_12) we can infer forall x_12: U_1(x_12) via universal simplification, which contradicts exists x_12: ~U_1(x_12). Therefore, the answer is no.