Consider the following premises. forall x_12: Q_12(x_12) -> P_12(x_12). q. ~Q_12(a_11). forall x_12: P_12(x_12) -> Q_12(x_12). Can we infer P_12(a_11) 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: 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(a_11) <-> Q_12(a_11) via universal instantiation. Finally, from P_12(a_11) <-> Q_12(a_11), ~Q_12(a_11) we can infer ~P_12(a_11) via biconditional elimination, which contradicts P_12(a_11). Therefore, the answer is no.