that's not quite how it works, unless your network has changed.
when I post on this site connected directly to my modem, my IP is
when my modem is connected to my router, and I connect to my router, the same outward facing IP is used.
when you connect to a router, the only change is that your computer is given an internal (router) IP as well.
If I post on my laptop from here, and my desktop, the same IP is shown to the mods.