Consider the following premises. forall x_5: T(x_5) -> Q_5(x_5). forall x_5: P_4(x_5) <-> Q_5(x_5). forall x_5: P_4(x_5). Can we infer exists x_5: ~Q_5(x_5) from them? If possible, name the inference rules being used at each step.
 From forall x_5: P_4(x_5) <-> Q_5(x_5) we can infer forall x_5: P_4(x_5) -> Q_5(x_5) via universal biconditional elimination. From forall x_5: P_4(x_5) we can infer forall x_5: P_4(x_5) or T(x_5) via universal addition. Finally, from forall x_5: P_4(x_5) -> Q_5(x_5), forall x_5: T(x_5) -> Q_5(x_5), forall x_5: P_4(x_5) or T(x_5) we can infer forall x_5: Q_5(x_5) via universal disjunction elimination, which contradicts exists x_5: ~Q_5(x_5). Therefore, the answer is no.