Consider the following premises. exists x: ~R(x). forall x: R(x) -> Q_1(x). forall x: Q_1(x) -> R(x). forall x: Q_1(x) <-> U(x). Can we infer forall x: U(x) from them? If possible, name the inference rules being used at each step.
 From forall x: Q_1(x) -> R(x), forall x: R(x) -> Q_1(x) we can infer forall x: Q_1(x) <-> R(x) via universal biconditional introduction. From forall x: Q_1(x) <-> R(x), exists x: ~R(x) we can infer exists x: ~Q_1(x) via existential biconditional elimination. Finally, from forall x: Q_1(x) <-> U(x), exists x: ~Q_1(x) we can infer exists x: ~U(x) via existential biconditional elimination, which contradicts forall x: U(x). Therefore, the answer is no.