Consider the following premises. forall z_3: ~Q_3(z_3) or P(z_3). forall z_3: Q_3(z_3) or U(z_3). forall z_3: R_4(z_3) -> S_4(z_3). exists z_3: ~U(z_3) or R_4(z_3). forall z_3: P(z_3) -> S_4(z_3). Can we infer exists z_3: S_4(z_3) from them? If possible, name the inference rules being used at each step.
 From forall z_3: Q_3(z_3) or U(z_3), forall z_3: ~Q_3(z_3) or P(z_3) we can infer forall z_3: U(z_3) or P(z_3) via universal resolution. From forall z_3: U(z_3) or P(z_3), exists z_3: ~U(z_3) or R_4(z_3) we can infer exists z_3: P(z_3) or R_4(z_3) via existential resolution. Finally, from forall z_3: P(z_3) -> S_4(z_3), forall z_3: R_4(z_3) -> S_4(z_3), exists z_3: P(z_3) or R_4(z_3) we can infer exists z_3: S_4(z_3) via existential disjunction elimination. Therefore, the answer is yes.