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.

$1.$ Identify the atomic propositions.

$2.$ Formalise the argument in propositional logic.

$3.$ Provide a formal, deductive proof of the validity of the argument. Every step of your proof must be numbered and fully annotated.