Consider the following premises. forall x_7: P(x_7) <-> S(x_7). forall x_7: S(x_7) and U(x_7). Can we infer forall x_7: P(x_7) from them?
 From forall x_7: S(x_7) and U(x_7) we can infer forall x_7: S(x_7). From forall x_7: P(x_7) <-> S(x_7) we can infer forall x_7: S(x_7) -> P(x_7). Finally, from forall x_7: S(x_7) -> P(x_7), forall x_7: S(x_7) we can infer forall x_7: P(x_7). Therefore, the answer is yes.