Consider the following premises. forall x_11: Q_3(x_11). forall x_11: P_11(x_11) <-> Q_11(x_11). forall x_11: Q_11(x_11) or Q_7(x_11). forall x_11: Q_7(x_11) -> P_11(x_11). Can we infer ~P_11(e) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_11(x_11) <-> Q_11(x_11) we can infer forall x_11: Q_11(x_11) -> P_11(x_11) via universal biconditional elimination. Finally, from forall x_11: Q_11(x_11) -> P_11(x_11), forall x_11: Q_7(x_11) -> P_11(x_11), forall x_11: Q_11(x_11) or Q_7(x_11) we can infer forall x_11: P_11(x_11) via universal disjunction elimination, which contradicts ~P_11(e). Therefore, the answer is no.