Consider the following premises. exists x: ~U(x). forall x: Q_1(x) <-> R(x). forall x: U(x) <-> Q_1(x). Can we infer exists x: ~R(x) from them? If possible, name the inference rules being used at each step.
 From forall x: U(x) <-> Q_1(x), exists x: ~U(x) we can infer exists x: ~Q_1(x) via existential biconditional elimination. Finally, from forall x: Q_1(x) <-> R(x), exists x: ~Q_1(x) we can infer exists x: ~R(x) via existential biconditional elimination. Therefore, the answer is yes.