Mark Wells wrote:
See http://www.woodbin.com/calcs/sagulator.htm

Gut tells me that two 1/2" sheets of ply glued together would be much
weaker than 1" thick ply.

Mark

I've read that a sheet of 3/4 ply is stronger than two 1/2 sheets glued
together. I can't prove it (lost the references), but it makes sense.

To the original poster. If it was me and I was dead set against having
a center support and it had to be 48" long.. I would use 3/4 ply, and
put a 1.5" strip of hardwood on the front. If you attach the shelves to
your 3/4 back, that will help too.

Another option is to just make the bookcase 32" wide. 3/4 ply
(particularly with a 1.5" edge) would definitely handle that span.