$4\mathrm{.}2$ Driver and Vehicle Licensing

A document contains the following axioms.

If you are the registered keeper of the vehicle and you have not paid tax on the vehicle then it is illegal for you to drive the vehicle.

If you do not have a driving licence or you are not insured to drive the vehicle then it is illegal for you to drive the vehicle.

If the vehicle is$,$ at least, three years old and does not have a valid MOT certificate, or you are not insured to drive it$,$ then you have not paid tax on the vehicle.

You are the registered keeper of the vehicle, have a driving licence and are insured to drive the vehicle.

The vehicle does not have a valid MOT certificate.

It is legal for you to drive the vehicle.

The document also contains the following conclusion.

Therefore, the vehicle is less than three years old.

Identify the atomic propositions.

Formalise the argument in propositional logic.

Provide a formal, deductive proof of the validity of the argument. Every step of your proof must be numbered and fully annotated.