Consider the following premises. forall x_12: Q_12(x_12) -> P_12(x_12). ~P_12(a_11). ~Q_12(a_11) -> s_1. forall x_12: P_12(x_12) -> Q_12(x_12). Can we infer ~s_1 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. From P_12(a_11) <-> Q_12(a_11), ~P_12(a_11) we can infer ~Q_12(a_11) via biconditional elimination. Finally, from ~Q_12(a_11) -> s_1, ~Q_12(a_11) we can infer s_1 via modus ponens, which contradicts ~s_1. Therefore, the answer is no.