Consider the following premises. forall x_16: P_7(x_16) <-> Q(x_16). forall x_16: P(x_16) -> P_2(x_16). forall x_16: P_16(x_16) <-> P_2(x_16). forall x_16: P_16(x_16) -> Q(x_16). forall x_16: ~P_7(x_16). Can we infer exists x_16: P(x_16) from them? If possible, name the inference rules being used at each step.
 From forall x_16: P_16(x_16) <-> P_2(x_16) we can infer forall x_16: P_2(x_16) -> P_16(x_16) via universal biconditional elimination. From forall x_16: P_2(x_16) -> P_16(x_16), forall x_16: P_16(x_16) -> Q(x_16) we can infer forall x_16: P_2(x_16) -> Q(x_16) via universal transitivity. From forall x_16: P_7(x_16) <-> Q(x_16), forall x_16: ~P_7(x_16) we can infer forall x_16: ~Q(x_16) via universal biconditional elimination. From forall x_16: P(x_16) -> P_2(x_16), forall x_16: P_2(x_16) -> Q(x_16) we can infer forall x_16: P(x_16) -> Q(x_16) via universal transitivity. Finally, from forall x_16: P(x_16) -> Q(x_16), forall x_16: ~Q(x_16) we can infer forall x_16: ~P(x_16) via universal modus tollens, which contradicts exists x_16: P(x_16). Therefore, the answer is no.