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