Consider the following premises. forall x: U(x) <-> R_1(x). forall x: R_1(x) -> P_1(x). forall x: U(x) <-> P_1(x). Can we infer exists x: ~(U(x) <-> P_1(x)) from them? If possible, name the inference rules being used at each step.
 From forall x: U(x) <-> R_1(x) we can infer forall x: U(x) -> R_1(x) via universal biconditional elimination. From forall x: U(x) <-> P_1(x) we can infer forall x: P_1(x) -> U(x) via universal biconditional elimination. From forall x: U(x) -> R_1(x), forall x: R_1(x) -> P_1(x) we can infer forall x: U(x) -> P_1(x) via universal transitivity. Finally, from forall x: U(x) -> P_1(x), forall x: P_1(x) -> U(x) we can infer forall x: U(x) <-> P_1(x) via universal biconditional introduction, which contradicts exists x: ~(U(x) <-> P_1(x)). Therefore, the answer is no.