If the purpose of laws is to contribute to people's happiness, we have a basis for criticizing existing laws as well as proposing new laws.
Main Conclusion (we must conclude)
Existing laws acquire legitimacy simply because they are laws.
Intermediate Conclusion (Hence)
If the purpose of laws is not to contribute to people's happiness, then we have no basis for the evaluation of existing laws.
Evidence
If the purpose of laws is to contribute to people's happiness, then have a basis for criticizing existing laws as well as proposing new laws.
Evaluate
Since the argument draws two conclusions, we can examine both moves. The move from the 1st claim to the 2nd claim commits the #1 famous flaw Necessary vs. Sufficient, in which the author introduces a conditional statement and then interprets it using illegal backwards or negated logic.
The first sentence says purpose is happiness → we have a basis The conclusion drawn from that is just an illegal negation. purpose is not happiness → do not have basis
The move from the 2nd claim to the 3rd claim is saying, .
We might be able to fight that by saying, or ?
Goal
It seems more likely that we're just being tested on the age old Necessary vs. Sufficient flaw and that the bonus conclusion was put in there just as a way to distract us and "hide the ball".
Whenever we're doing Flaw and see any conditional logic, we should immediately start suspecting an illegal reversal or illegal negation (aka, Necessary vs. Sufficient)