Consider the following premises. forall x_6: W(x_6). exists x_6: W(x_6) -> P_5(x_6). forall x_6: P_5(x_6) -> Q_6(x_6). forall x_6: Q_6(x_6) -> P_5(x_6). Can we infer forall x_6: ~P_5(x_6) from them? If possible, name the inference rules being used at each step.
 From forall x_6: W(x_6) we can infer forall x_6: W(x_6) or Q_6(x_6) via universal addition. From forall x_6: P_5(x_6) -> Q_6(x_6), forall x_6: Q_6(x_6) -> P_5(x_6) we can infer forall x_6: P_5(x_6) <-> Q_6(x_6) via universal biconditional introduction. From forall x_6: P_5(x_6) <-> Q_6(x_6) we can infer forall x_6: Q_6(x_6) -> P_5(x_6) via universal biconditional elimination. Finally, from exists x_6: W(x_6) -> P_5(x_6), forall x_6: Q_6(x_6) -> P_5(x_6), forall x_6: W(x_6) or Q_6(x_6) we can infer exists x_6: P_5(x_6) via existential disjunction elimination, which contradicts forall x_6: ~P_5(x_6). Therefore, the answer is no.