Consider the following premises. forall x_6: ~Q_6(x_6). forall x_6: P_6(x_6) -> Q_5(x_6). forall x_6: Q_5(x_6) -> P_6(x_6). forall x_6: P_6(x_6) -> Q_6(x_6). forall x_6: Q_6(x_6) -> P_6(x_6). Can we infer forall x_6: ~Q_5(x_6) from them? If possible, name the inference rules being used at each step.
 From forall x_6: P_6(x_6) -> Q_6(x_6), forall x_6: Q_6(x_6) -> P_6(x_6) we can infer forall x_6: P_6(x_6) <-> Q_6(x_6) via universal biconditional introduction. From forall x_6: P_6(x_6) -> Q_5(x_6), forall x_6: Q_5(x_6) -> P_6(x_6) we can infer forall x_6: P_6(x_6) <-> Q_5(x_6) via universal biconditional introduction. From forall x_6: P_6(x_6) <-> Q_6(x_6), forall x_6: ~Q_6(x_6) we can infer forall x_6: ~P_6(x_6) via universal biconditional elimination. Finally, from forall x_6: P_6(x_6) <-> Q_5(x_6), forall x_6: ~P_6(x_6) we can infer forall x_6: ~Q_5(x_6) via universal biconditional elimination. Therefore, the answer is yes.