Consider the following premises. forall x: Q(x) -> R(x). forall x: R_1(x) -> Q(x). exists x: R_1(x) or ~R(x). Can we infer exists x: R_1(x) <-> R(x) from them?
 From exists x: R_1(x) or ~R(x) we can infer exists x: R(x) -> R_1(x). From forall x: R_1(x) -> Q(x), forall x: Q(x) -> R(x) we can infer forall x: R_1(x) -> R(x). Finally, from forall x: R_1(x) -> R(x), exists x: R(x) -> R_1(x) we can infer exists x: R_1(x) <-> R(x). Therefore, the answer is yes.