Consider the following premises. forall x: W(x) -> Q(x). forall x: U_2(x) <-> Q(x). exists x: U_2(x) -> W(x). Can we infer exists x: U_2(x) <-> Q(x) from them?
 From exists x: U_2(x) -> W(x), forall x: W(x) -> Q(x) we can infer exists x: U_2(x) -> Q(x). From forall x: U_2(x) <-> Q(x) we can infer forall x: Q(x) -> U_2(x). Finally, from exists x: U_2(x) -> Q(x), forall x: Q(x) -> U_2(x) we can infer exists x: U_2(x) <-> Q(x). Therefore, the answer is yes.