Backup IMAP inbox and sent automatically to my mac's hard drive using rules

I have a 1and1.co.uk account setup for my emails. It only holds 2GB of storage. I have setup my iMac, iPad mini and iPhone5 mail accounts as IMAP to keep track of what I have sent and received.

I was hoping to create a back up on my iMac of sent and received emails so that I can regularly delete emails off the server to keep the 2GB available.I have set up some rules to copy emails received and sent that are older then 1 day to folders in Mac Mail under the "On My Mac" heading. I assumed that if the files are copied, then when I delete them off the 1and1 server they would not delete off my computer as they are stored under the "On My Mac" heading.

However this is not the case. When I delete them off the server they are deleted off my entire iMac,despite being copied to folders"On My Mac".

Please can someone tell me if "On My Mac" folders are still linked to your IMAP account and therefore still are affected if you delete items off the server? And if so, How can I autamatically or manually back up my Inbox and Sent items so thatthey are not affected when I delete off the server.