Consider the following premises. forall x_12: R_12(x_12) <-> S_12(x_12). forall x_10: ~R_10(x_10). S_12(a_12) -> R_10(a_10). Can we infer ~R_12(a_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: R_12(x_12) <-> S_12(x_12) we can infer R_12(a_12) <-> S_12(a_12) via universal instantiation. From forall x_10: ~R_10(x_10) we can infer ~R_10(a_10) via universal instantiation. From S_12(a_12) -> R_10(a_10), ~R_10(a_10) we can infer ~S_12(a_12) via modus tollens. Finally, from R_12(a_12) <-> S_12(a_12), ~S_12(a_12) we can infer ~R_12(a_12) via biconditional elimination. Therefore, the answer is yes.