It's a bit hard to follow what parts are given as input and what is part of the proof. For instance is (6) a proposition or a conclusion of yours? Can you post the exercise clearly separated from the solution?

Note: In real life it is true that a cat is and animal; in an excise though it could be part of the task to spot that the given information is insufficient so it is not possible to prove who killed the cat. Hence some clarity is required to allows for guidance in this case.