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