Consider the following premises. forall z_4: U_1(z_4) or ~Q(z_4). forall z_4: P_1(z_4) -> Q(z_4). exists z_4: P_1(z_4) or ~U_1(z_4). Can we infer forall z_4: ~(Q(z_4) <-> U_1(z_4)) from them? If possible, name the inference rules being used at each step.
 From exists z_4: P_1(z_4) or ~U_1(z_4) we can infer exists z_4: U_1(z_4) -> P_1(z_4) via existential rewriting. From exists z_4: U_1(z_4) -> P_1(z_4), forall z_4: P_1(z_4) -> Q(z_4) we can infer exists z_4: U_1(z_4) -> Q(z_4) via existential transitivity. From forall z_4: U_1(z_4) or ~Q(z_4) we can infer forall z_4: Q(z_4) -> U_1(z_4) via universal rewriting. Finally, from forall z_4: Q(z_4) -> U_1(z_4), exists z_4: U_1(z_4) -> Q(z_4) we can infer exists z_4: Q(z_4) <-> U_1(z_4) via existential biconditional introduction, which contradicts forall z_4: ~(Q(z_4) <-> U_1(z_4)). Therefore, the answer is no.