Most certainly you can paint over your existing paint without having to sand or prime. You've already done the hard work. The only problem I can foresee is with having the new "whiter" paint cover in one coat. The whiter the paint is the less tint in actually in the paint. Plus semi-gloss paints won't cover as well as flat or satin sheen paint.
One way of helping your new paint cover better is to have the paint store put a shot of white tint into the formula. This could change the color a bit, so talk to the sales person first. Another way is to apply a coat of exterior flat first, tinted to your new white. Then paint another coat with the semi-gloss.
Test the new white first before going through the trouble of painting with a flat first.