JS is right, you are qualified to weld using a specific code....typical when welding on bridges they will qualify the welders to AWS D1.5 structural welding code for bridges. For buildings and other things they may elect to use AWS D1.1 structural welding code for structural steel.
To be perfectly honest, there isn't much difference in the welding qualification portion of the test between the two codes(D1.5 only allows manual chipping of slag, no mechanical cleaning processes allowed). However for dual qualifications for both D1.1 and D1.5, the testing person needs to make certain that all of the variables for each code are being followed, for each code, at the time of testing.
Ultimately, as others have mentioned, it is up to the engineer who is responsible for reviewing submittals prior to the start of a job. If he says he will accept your existing certs then you are good to go. And, yes, for the most part your bridge code applicable certs would also be applicable for D1.1 Structural Steel. It would be nice if the testing company, person, would write everything up that way, test that way, and then fill out the paperwork with two separate certs so that you have no dispute with anyone when getting approved for a job.
But, bottomline in your particular instance, yes, if the boss says he wants you to weld on it, do it. While he is wrong about a "cert is a cert", he can use you on any job he wants. But when a job gets reviewed by the engineer they may want both certs instead of just the one. It's all up to them.
Good luck.
Have a Great Day, Brent