As promised, Feedly has pulled the trigger, removing its dependency on Google Reader and updating its iOS app to relfect the change. While it can still connect to Google Reader, Feedly is now officially powered by its own backend service, and will continue to function after Google Reader shuts down on July 1.
Many developers have been working on Google Reader replacements since the announcement of the shutdown, but Feedly is one of the first to launch their solution to the public. And it won't be restricted to their mobile and web apps, either. Popular RSS apps Reeder and Newsify have both said that they will offer access to Feedly before the end of this month, when Google Reader shuts down for good.
The Feedly update is currently available on the App Store.