Can the player manuals of the Essentials line (Heroes of the Forgotten Kingdoms / of the Fallen lands) replace the Player Handbooks?
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Short answer: No.
Long answer: The Heroes of the Forgotten Kingdoms and Fallen Lands books only cover the creation of characters, including races, classes, skills, feats, and gear. What they don't cover is the combat rules (second half of the Player's Handbook).
You have two choices here: Buy the Heroes books and the Rules Compendium to have all the rules you need to play. Or buy the Player's Handbook (the first one will probably suffice for starters). In both cases the rules will be the same. But in the Heroes books character creation has been simplified and has less options. The disadvantage of the Player's Handbook is that it doesn't contain all the errata.