Consider the following premises. forall x_6: P_6(x_6) -> U(x_6). forall x_6: U(x_6) -> P_6(x_6). forall x_6: U(x_6) -> P_6(x_6). Can we infer exists x_6: ~(U(x_6) -> P_6(x_6)) from them? If possible, name the inference rules being used at each step.
 From forall x_6: P_6(x_6) -> U(x_6), forall x_6: U(x_6) -> P_6(x_6) we can infer forall x_6: P_6(x_6) <-> U(x_6) via universal biconditional introduction. From forall x_6: P_6(x_6) <-> U(x_6) we can infer forall x_6: P_6(x_6) -> U(x_6) via universal biconditional elimination. From forall x_6: U(x_6) -> P_6(x_6), forall x_6: P_6(x_6) -> U(x_6) we can infer forall x_6: U(x_6) <-> P_6(x_6) via universal biconditional introduction. Finally, from forall x_6: U(x_6) <-> P_6(x_6) we can infer forall x_6: U(x_6) -> P_6(x_6) via universal biconditional elimination, which contradicts exists x_6: ~(U(x_6) -> P_6(x_6)). Therefore, the answer is no.