Consider the following premises. forall x: ~T(x). forall x: R(x) -> T(x). forall x: T(x) -> R(x). exists x: U(x) <-> R(x). Can we infer forall x: U(x) from them? If possible, name the inference rules being used at each step.
 From forall x: T(x) -> R(x), forall x: R(x) -> T(x) we can infer forall x: T(x) <-> R(x) via universal biconditional introduction. From forall x: T(x) <-> R(x), forall x: ~T(x) we can infer forall x: ~R(x) via universal biconditional elimination. Finally, from exists x: U(x) <-> R(x), forall x: ~R(x) we can infer exists x: ~U(x) via existential biconditional elimination, which contradicts forall x: U(x). Therefore, the answer is no.