You need to check with your wife's plan about coordinating the coverages. If her plan is secondary then you would submit your claim there after getting an EOB from the claim you make to your plan.
A simple example of how coordinated coverage works: if your plan covers 50% and hers covers 50%, then hers will pay nothing. If hers covers 60%, then hers will pay 10%. Of course this is not the only way it can work, but it's generally true.
If you thought you would get 50% back from yours and another 50% back from hers, that is you-know-what.
what you're referring to is "non-duplication of benefits".
Coordination of benefits is something else.
If your Delta pays 50% of the charges up to $2000, and your wife's Metlife pays 50% of the charges up to $2000, and your bill is $4000, Delta will pay half, then Metlife will pay half of the half...and your portion is 25%.
If your bill is $5000, then Delta will pay $2000, and Metlife will pay $1500, and your portion is $1500.
Does that make sense?
Make sure Delta is primary. If it is secondary, it frequently has non-duplication of benefits, in which case it will pay nothing.