Consider the following premises. forall x_8: P_4(x_8) -> Q_8(x_8). forall x_8: Q_8(x_8) -> P_4(x_8). Q_8(e) <-> w_4. ~P_4(e). Can we infer ~w_4 from them?
 From forall x_8: P_4(x_8) -> Q_8(x_8), forall x_8: Q_8(x_8) -> P_4(x_8) we can infer forall x_8: P_4(x_8) <-> Q_8(x_8). From forall x_8: P_4(x_8) <-> Q_8(x_8) we can infer P_4(e) <-> Q_8(e). From P_4(e) <-> Q_8(e), ~P_4(e) we can infer ~Q_8(e). Finally, from Q_8(e) <-> w_4, ~Q_8(e) we can infer ~w_4. Therefore, the answer is yes.