Dear Bronx Tenant:
The landlord may charge rent only as set out in the lease. If the landlord could not deliver possession until July 15, 2014, the lease should provide that rent is due only from the date the landlord delivered vacant possession into a ready to move in apartment. New York law does not allow a double-dip for the landlord in rent collection. Since the prior tenant paid July, none of your first month rent should have been charged to July. Your first month rent would cover August and you would next pay rent September 1, 2014.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.