Consider the following premises. forall x: ~P(x). forall x: P(x) -> R(x). exists x: P(x) or ~R(x). Can we infer exists x: ~R(x) from them?
 From exists x: P(x) or ~R(x) we can infer exists x: R(x) -> P(x). From forall x: P(x) -> R(x), exists x: R(x) -> P(x) we can infer exists x: P(x) <-> R(x). Finally, from exists x: P(x) <-> R(x), forall x: ~P(x) we can infer exists x: ~R(x). Therefore, the answer is yes.