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