Consider the following premises. forall x_6: Q_6(x_6) -> U(x_6). forall x_6: P_2(x_6) -> P_6(x_6). exists x_6: P_6(x_6) -> Q_6(x_6). forall x_6: U(x_6) -> P_2(x_6). Can we infer forall x_6: ~(U(x_6) <-> P_2(x_6)) from them? If possible, name the inference rules being used at each step.
 From exists x_6: P_6(x_6) -> Q_6(x_6), forall x_6: Q_6(x_6) -> U(x_6) we can infer exists x_6: P_6(x_6) -> U(x_6) via existential transitivity. From forall x_6: P_2(x_6) -> P_6(x_6), exists x_6: P_6(x_6) -> U(x_6) we can infer exists x_6: P_2(x_6) -> U(x_6) via existential transitivity. Finally, from forall x_6: U(x_6) -> P_2(x_6), exists x_6: P_2(x_6) -> U(x_6) we can infer exists x_6: U(x_6) <-> P_2(x_6) via existential biconditional introduction, which contradicts forall x_6: ~(U(x_6) <-> P_2(x_6)). Therefore, the answer is no.