Consider the following premises. forall y: Q_4(y) <-> R(y). forall y: R(y) or ~Q_4(y). Can we infer exists y: ~(R(y) <-> Q_4(y)) from them? If possible, name the inference rules being used at each step.
 From forall y: Q_4(y) <-> R(y) we can infer forall y: R(y) -> Q_4(y) via universal biconditional elimination. From forall y: R(y) or ~Q_4(y) we can infer forall y: Q_4(y) -> R(y) via universal rewriting. Finally, from forall y: R(y) -> Q_4(y), forall y: Q_4(y) -> R(y) we can infer forall y: R(y) <-> Q_4(y) via universal biconditional introduction, which contradicts exists y: ~(R(y) <-> Q_4(y)). Therefore, the answer is no.