Consider the following premises. ~Q_12(e). s_1 and u_1. forall x_12: Q_12(x_12) -> P_12(x_12). forall x_12: P_12(x_12) -> Q_12(x_12). Can we infer P_12(e) from them? If possible, name the inference rules being used at each step.
 From s_1 and u_1 we can infer s_1 via simplification. From forall x_12: P_12(x_12) -> Q_12(x_12), forall x_12: Q_12(x_12) -> P_12(x_12) we can infer forall x_12: P_12(x_12) <-> Q_12(x_12) via universal biconditional introduction. From forall x_12: P_12(x_12) <-> Q_12(x_12) we can infer P_12(e) <-> Q_12(e) via universal instantiation. Finally, from P_12(e) <-> Q_12(e), ~Q_12(e) we can infer ~P_12(e) via biconditional elimination, which contradicts P_12(e). Therefore, the answer is no.