Consider the following premises. forall x_10: Q_3(x_10). forall x_10: Q_7(x_10) -> Q_10(x_10). forall x_10: P_10(x_10) or Q_7(x_10). forall x_10: P_10(x_10) <-> Q_10(x_10). Can we infer ~Q_10(a_2) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_10(x_10) <-> Q_10(x_10) we can infer forall x_10: P_10(x_10) -> Q_10(x_10) via universal biconditional elimination. Finally, from forall x_10: P_10(x_10) -> Q_10(x_10), forall x_10: Q_7(x_10) -> Q_10(x_10), forall x_10: P_10(x_10) or Q_7(x_10) we can infer forall x_10: Q_10(x_10) via universal disjunction elimination, which contradicts ~Q_10(a_2). Therefore, the answer is no.