Consider the following premises. forall z: ~P_4(z). forall z: S_1(z) -> P(z). forall z: P(z) -> P_4(z). exists z: S_1(z) or ~P_4(z). Can we infer exists z: ~S_1(z) from them? If possible, name the inference rules being used at each step.
 From exists z: S_1(z) or ~P_4(z) we can infer exists z: P_4(z) -> S_1(z) via existential rewriting. From forall z: S_1(z) -> P(z), forall z: P(z) -> P_4(z) we can infer forall z: S_1(z) -> P_4(z) via universal transitivity. From exists z: P_4(z) -> S_1(z), forall z: S_1(z) -> P_4(z) we can infer exists z: P_4(z) <-> S_1(z) via existential biconditional introduction. Finally, from exists z: P_4(z) <-> S_1(z), forall z: ~P_4(z) we can infer exists z: ~S_1(z) via existential biconditional elimination. Therefore, the answer is yes.