What city? I would bet hard money that you can't find that in writing. I pull out atmospheric vacuum breakers all the time, because nowhere on the planet Earth are they approved for backflow protection with any valves downstream of them. You can ask the manufacturers, and they will tell you the same.
Understand this about the plumbing codes - your city does not make the rules. Your state has a building code, and the building code will refer to one or another regional plumbing codes. California wrote their own rules, and refers to just about nobody. The only thing your town or city can do is set tougher rules than the regional codes. One such is when a town will absolutely forbid the use of atmospheric vacuum breakers.
Spend some time at your local library, and ask your reference desk for the code book. There are about three or four valid forms of lawn sprinkler system backflow prevention, and you will see them spelled out in the code. A dual check valve has no place in a sprinkler system, because it cannot pass the very simple requirement of <b>"Prove to me that it works."</b> - that's where the codes set the bar. If you still are uncertain about this, go and ask the town official that told you that a dual check and a vacuum breaker was good enough to put it in writing on official letterhead, and to sign and date it.
If you are going to do a proper job, you are going to have to change your design or upgrade your components.