Here is the argument:
No N is not-N.
No not-N is N.
All C are N.
No R are C.
Thus: No R are N.
The syllogism is invalid for two reasons. First, the third premise denies the antecedent (cats) of the fourth. There can be other animals that are normal. Wikipedia: Denying the antecedent; Formal fallacyDenying the antecedent; Formal fallacy.
Second, a term that is distributed in the conclusion (normal) is not distributed in the major premise (all cats are normal). Wikipedia: Illicit majorIllicit major.
The first two premises are not needed except as definitions. The second two, about cats and rabbits, state actual relationships between categories,