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