Consider the following premises. forall x: ~Q_1(x). forall x: P(x) -> Q_1(x). forall x: Q_1(x) -> P(x). forall x: P(x) -> Q_1(x). Can we infer exists x: P(x) from them?
 From forall x: Q_1(x) -> P(x), forall x: P(x) -> Q_1(x) we can infer forall x: Q_1(x) <-> P(x). From forall x: Q_1(x) <-> P(x) we can infer forall x: Q_1(x) -> P(x). From forall x: Q_1(x) -> P(x), forall x: P(x) -> Q_1(x) we can infer forall x: Q_1(x) <-> P(x). Finally, from forall x: Q_1(x) <-> P(x), forall x: ~Q_1(x) we can infer forall x: ~P(x), which contradicts exists x: P(x). Therefore, the answer is no.