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Check the harness to the Ignition switch. As the contacts get old they can come loose. The switch may be bad, but I would check the plug before replacing anything. The clue is your hitting the brake and causing a change. The harness runs down the top of the steering column and then right above the brake pedal. Your hitting the brakes may be shaking this wire. I cannot think of anything under the hood that would cause this when you hit the brakes othe than a loose ground wire. Good Luck.