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