Do you mean, so you can use SSIs without changing the URLs? Sure, you could do that, just like you could rewrite URLs in .html to serve up .php pages. But, unlike the php analogy (which I thought of because I actually do it here and there), there are alternatives. The simplest is just to say
AddOutputFilter INCLUDES .html .php
Or, if only a few specific pages are involved, there's the X-Bit Hack, which is less scary than it sounds.
But wait. Will your pages no longer be using php content? Otherwise you'd need to throw in a different AddOutputFilter line to parse all your shtml pages for php, and then what was the point of changing? If you're already in php, why not just use php includes for everything?
You said "rewrite" but I can't help noticing that the second URL is "prettier" than the first one. Did you really mean an external redirect as part of changing your whole URL structure? At this point, some people would be encouraging you to think about going extensionless. My personal reaction to an extensionless URL is "Go back in the server and put some clothes on!" but this is, I think, a minority opinion.