Consider the following premises. exists y: ~R(y). forall y: U_1(y) -> R(y). forall y: R(y) -> U_1(y). Can we infer forall y: U_1(y) from them? If possible, name the inference rules being used at each step.
 From forall y: U_1(y) -> R(y), forall y: R(y) -> U_1(y) we can infer forall y: U_1(y) <-> R(y) via universal biconditional introduction. Finally, from forall y: U_1(y) <-> R(y), exists y: ~R(y) we can infer exists y: ~U_1(y) via existential biconditional elimination, which contradicts forall y: U_1(y). Therefore, the answer is no.