Consider the following premises. forall x: U_3(x) -> Q_4(x). forall x: U_3(x) -> Q_4(x). forall x: Q_4(x) -> U_3(x). Can we infer exists x: ~(Q_4(x) <-> U_3(x)) from them? If possible, name the inference rules being used at each step.
 From forall x: U_3(x) -> Q_4(x), forall x: Q_4(x) -> U_3(x) we can infer forall x: U_3(x) <-> Q_4(x) via universal biconditional introduction. From forall x: U_3(x) <-> Q_4(x) we can infer forall x: Q_4(x) -> U_3(x) via universal biconditional elimination. Finally, from forall x: Q_4(x) -> U_3(x), forall x: U_3(x) -> Q_4(x) we can infer forall x: Q_4(x) <-> U_3(x) via universal biconditional introduction, which contradicts exists x: ~(Q_4(x) <-> U_3(x)). Therefore, the answer is no.