In short, no. The lender has to tell you that they will perform a search as per their account opening procedures, but I do not think they are obliged to tell you which one they will use.
Generally speaking, lenders use both of them to get a bigger picture of the potential customer. If there is any information held with Equifax which you feel should not be there (thus affecting your credit history), you should get in contact with them and get any adverse data removed.
I hope that this helps, and good luck!!