They probably won't last as long as the Bridgestone's. Not much else will last as long, the Bridgestone's are quite hard tyres, which is why they're noisy and have less grip.
If you're planning to keep the car a while longer, it may be worth swapping tyres round so the older ones go on the front to balance out the wear, and put some new Michelin's on the rear. Most tyres are past their best after 6 years anyway, whether they're worn or not.
If you put Michelin's on the front you'll get better traction pulling away from junctions. Ideally you'd have the same tyres all round, but if not it's advisable not to have different tyres on the same axle.