Consider the following premises. forall x: P(x) or ~S(x). exists x: ~S(x). forall x: P(x) <-> S(x). Can we infer forall x: P(x) from them?
 From forall x: P(x) <-> S(x) we can infer forall x: P(x) -> S(x). From forall x: P(x) or ~S(x) we can infer forall x: S(x) -> P(x). From forall x: S(x) -> P(x), forall x: P(x) -> S(x) we can infer forall x: S(x) <-> P(x). Finally, from forall x: S(x) <-> P(x), exists x: ~S(x) we can infer exists x: ~P(x), which contradicts forall x: P(x). Therefore, the answer is no.