Consider the following premises. forall z_4: U(z_4) -> Q_4(z_4). forall z_4: Q_4(z_4) -> U(z_4). ~U(e). w. Can we infer w and ~Q_4(e) from them? If possible, name the inference rules being used at each step.
 From forall z_4: Q_4(z_4) -> U(z_4), forall z_4: U(z_4) -> Q_4(z_4) we can infer forall z_4: Q_4(z_4) <-> U(z_4) via universal biconditional introduction. From forall z_4: Q_4(z_4) <-> U(z_4) we can infer Q_4(e) <-> U(e) via universal instantiation. From Q_4(e) <-> U(e), ~U(e) we can infer ~Q_4(e) via biconditional elimination. Finally, from w, ~Q_4(e) we can infer w and ~Q_4(e) via conjunction. Therefore, the answer is yes.