Consider the following premises. forall x: T(x) -> W(x). forall x: W(x) -> T(x). exists x: U(x) or ~W(x). Can we infer exists x: T(x) -> U(x) from them? If possible, name the inference rules being used at each step.
 From exists x: U(x) or ~W(x) we can infer exists x: W(x) -> U(x) via existential rewriting. From forall x: W(x) -> T(x), forall x: T(x) -> W(x) we can infer forall x: W(x) <-> T(x) via universal biconditional introduction. From forall x: W(x) <-> T(x) we can infer forall x: T(x) -> W(x) via universal biconditional elimination. Finally, from forall x: T(x) -> W(x), exists x: W(x) -> U(x) we can infer exists x: T(x) -> U(x) via existential transitivity. Therefore, the answer is yes.