Consider the following premises. forall x_8: P_8(x_8) -> Q_8(x_8). forall x_8: Q_8(x_8) -> P_8(x_8). P_8(b). Can we infer ~Q_8(b) from them? If possible, name the inference rules being used at each step.
 From forall x_8: P_8(x_8) -> Q_8(x_8), forall x_8: Q_8(x_8) -> P_8(x_8) we can infer forall x_8: P_8(x_8) <-> Q_8(x_8) via universal biconditional introduction. From forall x_8: P_8(x_8) <-> Q_8(x_8) we can infer P_8(b) <-> Q_8(b) via universal instantiation. From P_8(b) <-> Q_8(b) we can infer P_8(b) -> Q_8(b) via biconditional elimination. Finally, from P_8(b) -> Q_8(b), P_8(b) we can infer Q_8(b) via modus ponens, which contradicts ~Q_8(b). Therefore, the answer is no.