Consider the following premises. exists x_10: Q_6(x_10) -> Q(x_10). forall x_10: ~Q_10(x_10). forall x_10: P_10(x_10) <-> Q_10(x_10). forall x_10: Q(x_10) -> Q_6(x_10). forall x_10: Q(x_10) <-> P_10(x_10). Can we infer exists x_10: ~Q_6(x_10) 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), forall x_10: ~Q_10(x_10) we can infer forall x_10: ~P_10(x_10) via universal biconditional elimination. From forall x_10: Q(x_10) -> Q_6(x_10), exists x_10: Q_6(x_10) -> Q(x_10) we can infer exists x_10: Q(x_10) <-> Q_6(x_10) via existential biconditional introduction. From forall x_10: Q(x_10) <-> P_10(x_10), forall x_10: ~P_10(x_10) we can infer forall x_10: ~Q(x_10) via universal biconditional elimination. Finally, from exists x_10: Q(x_10) <-> Q_6(x_10), forall x_10: ~Q(x_10) we can infer exists x_10: ~Q_6(x_10) via existential biconditional elimination. Therefore, the answer is yes.