Consider the following premises. forall x_12: S_1(x_12) <-> Q_8(x_12). exists x_12: ~Q_8(x_12). forall x_12: R(x_12) -> S_1(x_12). forall x_12: S_1(x_12) <-> R(x_12). Can we infer forall x_12: R(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: S_1(x_12) <-> R(x_12) we can infer forall x_12: S_1(x_12) -> R(x_12) via universal biconditional elimination. From forall x_12: S_1(x_12) -> R(x_12), forall x_12: R(x_12) -> S_1(x_12) we can infer forall x_12: S_1(x_12) <-> R(x_12) via universal biconditional introduction. From forall x_12: S_1(x_12) <-> Q_8(x_12), exists x_12: ~Q_8(x_12) we can infer exists x_12: ~S_1(x_12) via existential biconditional elimination. Finally, from forall x_12: S_1(x_12) <-> R(x_12), exists x_12: ~S_1(x_12) we can infer exists x_12: ~R(x_12) via existential biconditional elimination, which contradicts forall x_12: R(x_12). Therefore, the answer is no.