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