Technically they are separate issues and separate disciplinary tracks, but there is a cumulative and compounding effect that could “fast track” the second disciplinary action from a warning to a written warning. We wouldn’t advise skipping more than 1 or 2 steps in your process, but that really depends on the severity of the performance issues.
We would also recommend amending your disciplinary process to spell out, if you are already on a PIP and are placed on another for a separate reason you immediately start with a written warning.
Without knowing the details, it would seem like you have a problem employee.