Strength of plywood walls for built-in bookshelves

I am planning on building built-in bookshelves that would span the length of a wall. Each unit would consist of a cabinet base ~2′ across with a set of bookshelves mounted on top. The depth of the cabinets would be ~2′ and the depth of each bookshelf would be ~12″.

The plan is to construct the cabinet and bookshelf carcasses out of 3/4″ plywood, then add hardwood face frames to each unit. 1/4″ (or 3/8″) rabbets would be cut for each shelf. Also, the cabinet and bookshelf would have plywood backings mounted, probably 1/2″. Everything would then be mounted to the wall and each other.

Is the strength of the plywood enough to hold up the weight of the bookshelves + books without bowing? I would like to avoid it, but should I go with additional upright 2x4s between each unit for added support?