Consider the following premises. P(e) -> Q_8(e). forall x_8: Q_8(x_8) -> P(x_8). forall x_8: P(x_8) -> Q_8(x_8). Can we infer Q_8(e) <-> P(e) from them? If possible, name the inference rules being used at each step.
 From forall x_8: P(x_8) -> Q_8(x_8), forall x_8: Q_8(x_8) -> P(x_8) we can infer forall x_8: P(x_8) <-> Q_8(x_8) via universal biconditional introduction. From forall x_8: P(x_8) <-> Q_8(x_8) we can infer P(e) <-> Q_8(e) via universal instantiation. From P(e) <-> Q_8(e) we can infer Q_8(e) -> P(e) via biconditional elimination. Finally, from Q_8(e) -> P(e), P(e) -> Q_8(e) we can infer Q_8(e) <-> P(e) via biconditional introduction. Therefore, the answer is yes.