Consider the following premises. forall y: R(y) -> T(y). forall y: R_3(y) -> U(y). exists y: R(y) or R_3(y). forall y: U(y) <-> T(y). Can we infer exists y: U(y) from them? If possible, name the inference rules being used at each step.
 From forall y: U(y) <-> T(y) we can infer forall y: T(y) -> U(y) via universal biconditional elimination. From forall y: R(y) -> T(y), forall y: T(y) -> U(y) we can infer forall y: R(y) -> U(y) via universal transitivity. Finally, from forall y: R(y) -> U(y), forall y: R_3(y) -> U(y), exists y: R(y) or R_3(y) we can infer exists y: U(y) via existential disjunction elimination. Therefore, the answer is yes.