Consider the following premises. forall x_7: P_7(x_7) <-> W(x_7). forall x_7: P_7(x_7) or ~W(x_7). Can we infer exists x_7: ~(W(x_7) <-> P_7(x_7)) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_7(x_7) <-> W(x_7) we can infer forall x_7: P_7(x_7) -> W(x_7) via universal biconditional elimination. From forall x_7: P_7(x_7) or ~W(x_7) we can infer forall x_7: W(x_7) -> P_7(x_7) via universal rewriting. Finally, from forall x_7: W(x_7) -> P_7(x_7), forall x_7: P_7(x_7) -> W(x_7) we can infer forall x_7: W(x_7) <-> P_7(x_7) via universal biconditional introduction, which contradicts exists x_7: ~(W(x_7) <-> P_7(x_7)). Therefore, the answer is no.