The PS4’s secure processor tracks the version number of the installed software. If a user attempts to install an update file with a lower version number than the current system, the console rejects it immediately.

The PS4 motherboard contains two "slots" for firmware: the current version and the previous version. If you recently updated to 13.02, your console still holds the data for your previous firmware (e.g., 12.52 or 12.00) in its backup slot.