Consider the following premises. forall x_9: P(x_9). Q_9(a_8) -> p_4. p_4 <-> P(a_8). Can we infer ~p_4 from them? If possible, name the inference rules being used at each step.
 From forall x_9: P(x_9) we can infer forall x_9: P(x_9) or Q_9(x_9) via universal addition. From forall x_9: P(x_9) or Q_9(x_9) we can infer P(a_8) or Q_9(a_8) via universal instantiation. From p_4 <-> P(a_8) we can infer P(a_8) -> p_4 via biconditional elimination. Finally, from P(a_8) -> p_4, Q_9(a_8) -> p_4, P(a_8) or Q_9(a_8) we can infer p_4 via disjunction elimination, which contradicts ~p_4. Therefore, the answer is no.