Consider the following premises. forall x: R(x) -> U_4(x). exists x: P_4(x) <-> U_2(x). forall x: U_4(x) -> P_4(x). forall x: R(x) or ~U_2(x). Can we infer exists x: U_2(x) <-> P_4(x) from them? If possible, name the inference rules being used at each step.
 From forall x: R(x) or ~U_2(x) we can infer forall x: U_2(x) -> R(x) via universal rewriting. From forall x: U_2(x) -> R(x), forall x: R(x) -> U_4(x) we can infer forall x: U_2(x) -> U_4(x) via universal transitivity. From exists x: P_4(x) <-> U_2(x) we can infer exists x: P_4(x) -> U_2(x) via existential biconditional elimination. From forall x: U_2(x) -> U_4(x), forall x: U_4(x) -> P_4(x) we can infer forall x: U_2(x) -> P_4(x) via universal transitivity. Finally, from forall x: U_2(x) -> P_4(x), exists x: P_4(x) -> U_2(x) we can infer exists x: U_2(x) <-> P_4(x) via existential biconditional introduction. Therefore, the answer is yes.