Consider the following premises. exists x: U(x) <-> R_1(x). forall x: U(x) -> U_2(x). forall x: U_2(x) -> R_1(x). Can we infer forall x: ~(U(x) <-> R_1(x)) from them? If possible, name the inference rules being used at each step.
 From exists x: U(x) <-> R_1(x) we can infer exists x: R_1(x) -> U(x) via existential biconditional elimination. From forall x: U(x) -> U_2(x), forall x: U_2(x) -> R_1(x) we can infer forall x: U(x) -> R_1(x) via universal transitivity. Finally, from forall x: U(x) -> R_1(x), exists x: R_1(x) -> U(x) we can infer exists x: U(x) <-> R_1(x) via existential biconditional introduction, which contradicts forall x: ~(U(x) <-> R_1(x)). Therefore, the answer is no.