The DM also needs the PHB for DMing. He needs to know what the PCs can and can't do and there is other important information that is critical to running a game that is only found in the PHB.
Now that you mention it, that's actually been a little less true of 4e & Essentials. Essentials, of course, with all it's redundancies, including the very handy RC. But just, in general, with 4e the DM has to know the rules, but doesn't really need to know classes inside-out to run a decent game. PC powers and features use the same rules, jargon, conditions, keywords and so forth as monsters, and, with the questionable wonder that is exception based design, you at least can look at each power in a vacuum and adjudicate how it's resolved pretty easily. There's just little need for the full-scale character-sheet auditing you might do in some other games.
Love 4e? Concerned about its future?
Join the Old Guard of 4e"You want The Tooth?
You can't handle The Tooth!" - Dahlver-Nar.
"If magic is unrestrained in the campaign, D&D quickly degenerates into a weird wizard show where players get bored quickly" - E. Gary Gygax
