Gmail applies all filters to all incoming mail (rather than in a specific order like most desktop mail clients). Therefore, You will need to create an additional filter that is the logical
NOT of all your other filters. For instance, let’s suppose you have the following filters in place:
from:ebay.com put in folder eBay
from:facebookmail.com put in folder Facebook
Your new forwarding rule would be:
-from:ebay.com AND -from:facebookmail.com forward to email@example.com
This can prove very tedious and error-prone, unfortunately, especially if you have many filters to invert, and even more so if they use complex rules. Here are a few alternative solutions that may or may not be better, depending on your exact circumstances:
- Forward all your email to firstname.lastname@example.org (in this example) and setup folder rules there.
- Ask your friends to email you at your email@example.com address.
- Setup your firstname.lastname@example.org address to fetch your Gmail email via POP.
To me the best option (and the one I believe best fits your conundrum) is through door #3, provided your email provider (senders.com in this example) supports accessing external accounts using POP. Most email providers do, and it’s very easy to setup:
- Click the Gear icon in your Gmail account
- Open your Settings.
- Navigate to the Forwarding and POP/IMAP tab.
- Tick either Enable POP for all mail (even mail that's already been downloaded) or Enable POP for mail that arrives from now on, as the case may be.
- Under the heading When messages are accessed with POP select delete Gmail’s copy.
- Save Changes.
- Open the webmail page for your senders.com email address.
- Go into its Settings page.
- Find a section titled External Accounts, or perhaps POP Accounts.
- Add your Gmail account using the following settings:
POP server: pop.gmail.com
Use SSL: Yes
Username: Your @gmail.com address
Password: Your password
If you’ve setup two-factor authentication for your Gmail account you will need to generate an application password, and use it instead for POP retrieval.
The biggest downside to this is you cannot control how often Gmail fetches POP mail. From my own observations as of June 2014, this is ≈ every 70mn. Old tricks to speed this up no longer work since ≈ 2013.