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