Consider the following premises. forall x: W_2(x) -> R_1(x). forall x: R_1(x) -> U_1(x). forall x: W_2(x) <-> U_1(x). forall x: ~W_2(x). exists x: U(x) <-> U_1(x). Can we infer exists x: ~U(x) from them? If possible, name the inference rules being used at each step.
 From forall x: W_2(x) <-> U_1(x) we can infer forall x: U_1(x) -> W_2(x) via universal biconditional elimination. From forall x: W_2(x) -> R_1(x), forall x: R_1(x) -> U_1(x) we can infer forall x: W_2(x) -> U_1(x) via universal transitivity. From forall x: U_1(x) -> W_2(x), forall x: W_2(x) -> U_1(x) we can infer forall x: U_1(x) <-> W_2(x) via universal biconditional introduction. From forall x: U_1(x) <-> W_2(x), forall x: ~W_2(x) we can infer forall x: ~U_1(x) via universal biconditional elimination. Finally, from exists x: U(x) <-> U_1(x), forall x: ~U_1(x) we can infer exists x: ~U(x) via existential biconditional elimination. Therefore, the answer is yes.