Consider the following premises. forall x: T(x) -> R_3(x). forall x: ~T(x). forall x: R_3(x) -> T(x). Can we infer exists x: R_3(x) from them? If possible, name the inference rules being used at each step.
 From forall x: R_3(x) -> T(x), forall x: T(x) -> R_3(x) we can infer forall x: R_3(x) <-> T(x) via universal biconditional introduction. Finally, from forall x: R_3(x) <-> T(x), forall x: ~T(x) we can infer forall x: ~R_3(x) via universal biconditional elimination, which contradicts exists x: R_3(x). Therefore, the answer is no.