Consider the following premises. forall x_10: Q_10(x_10). forall x_10: R(x_10) <-> W(x_10). forall x_10: R(x_10) -> U(x_10). forall x_10: U(x_10) -> ~Q_10(x_10). forall x_10: W(x_10) <-> R_4(x_10). Can we infer forall x_10: ~R_4(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: U(x_10) -> ~Q_10(x_10), forall x_10: Q_10(x_10) we can infer forall x_10: ~U(x_10) via universal modus tollens. From forall x_10: R(x_10) -> U(x_10), forall x_10: ~U(x_10) we can infer forall x_10: ~R(x_10) via universal modus tollens. From forall x_10: R(x_10) <-> W(x_10), forall x_10: ~R(x_10) we can infer forall x_10: ~W(x_10) via universal biconditional elimination. Finally, from forall x_10: W(x_10) <-> R_4(x_10), forall x_10: ~W(x_10) we can infer forall x_10: ~R_4(x_10) via universal biconditional elimination. Therefore, the answer is yes.