Hello and thank you for writing. The FHA does NOT allow mortgage insurance cancellation with a new appraisal. You have to pay the loan down to less than 78 percent of the original value / purchase price.
Because you obtained your FHA mortgage before June 3, 2013, you can cancel your mortgage insurance. But not by paying for a new appraisal.
It's probably smart to run the numbers and see what you could save with a refinance to a conventional (non-government) loan. Because otherwise, on a regular amortization schedule, you've probably got another five years of MIP if you don't.
Here's everything you need to know about canceling your FHA MIP now, regardless of when you bought your home. https://themortgagereports.com/7570/fha-mip-cancel