Assuming that; as long as it is true that; if it is the case that. I don't care if you wear headphones while you work, provided that it doesn't affect your productivity. I'd be happy to give you the information you're looking for, provided I get something in return, of course.
See also: provided
Farlex Dictionary of Idioms. © 2015 Farlex, Inc, all rights reserved.
on the condition that. I will come, provided that I am invited. Iwill help you, provided that you pay me.
McGraw-Hill Dictionary of American Idioms and Phrasal Verbs. © 2002 by The McGraw-Hill Companies, Inc.