Consider the following premises. exists z_2: Q(z_2) or ~S_3(z_2). forall z_2: Q(z_2) <-> R(z_2). forall z_2: R(z_2) or S_3(z_2). Can we infer exists z_2: Q(z_2) from them? If possible, name the inference rules being used at each step.
 From exists z_2: Q(z_2) or ~S_3(z_2) we can infer exists z_2: S_3(z_2) -> Q(z_2) via existential rewriting. From forall z_2: Q(z_2) <-> R(z_2) we can infer forall z_2: R(z_2) -> Q(z_2) via universal biconditional elimination. Finally, from forall z_2: R(z_2) -> Q(z_2), exists z_2: S_3(z_2) -> Q(z_2), forall z_2: R(z_2) or S_3(z_2) we can infer exists z_2: Q(z_2) via existential disjunction elimination. Therefore, the answer is yes.