Consider the following premises. ~R_6(a_5). forall x_12: Q_12(x_12) -> R_6(x_12). forall x_12: P_12(x_12) <-> Q_12(x_12). Can we infer ~P_12(a_5) 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) we can infer forall x_12: P_12(x_12) -> Q_12(x_12) via universal biconditional elimination. From forall x_12: P_12(x_12) -> Q_12(x_12), forall x_12: Q_12(x_12) -> R_6(x_12) we can infer forall x_12: P_12(x_12) -> R_6(x_12) via universal transitivity. From forall x_12: P_12(x_12) -> R_6(x_12) we can infer P_12(a_5) -> R_6(a_5) via universal instantiation. Finally, from P_12(a_5) -> R_6(a_5), ~R_6(a_5) we can infer ~P_12(a_5) via modus tollens. Therefore, the answer is yes.