Consider the following premises. forall x_5: R(x_5) -> P_5(x_5). exists x_5: Q_4(x_5) <-> R(x_5). forall x_5: ~P_5(x_5). forall x_5: P_5(x_5) -> R(x_5). Can we infer exists x_5: ~Q_4(x_5) from them? If possible, name the inference rules being used at each step.
 From forall x_5: P_5(x_5) -> R(x_5), forall x_5: R(x_5) -> P_5(x_5) we can infer forall x_5: P_5(x_5) <-> R(x_5) via universal biconditional introduction. From forall x_5: P_5(x_5) <-> R(x_5), forall x_5: ~P_5(x_5) we can infer forall x_5: ~R(x_5) via universal biconditional elimination. Finally, from exists x_5: Q_4(x_5) <-> R(x_5), forall x_5: ~R(x_5) we can infer exists x_5: ~Q_4(x_5) via existential biconditional elimination. Therefore, the answer is yes.