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