The brake pedal feels solid but as soon as the car is started the pressure drops

My brake pedal has been low for several months. I took it in and the mechanics put in new rear brakes. The brake pedal was still low. They bled the brakes several different times. They put in a new master cylinder. The brake pedal was still low. They put in a new proportioning valve. The brake pedal is still low. It goes almost to the floor before I feel any braking and if I press hard enough when stopped I can push it to the floor. The brake pedal feels solid enough when the car is not started but as soon as the car is started the pressure drops. The mechanics have tried everything they know. The bled the lines multiple times even when the car was jacked up. They checked the calipers, did something with a wheel cylinder, it seems like they did everything, except fix it. Anyway I am 700.00 out and have the same problem as before I went in. I don't know what to do or where to go next?
Please help