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