Can you tell me which word is equivalent to providing in this sentence below, if or in case?
The deposit is refundable when you leave, providing you haven't damaged anything.
Can you tell me which word is equivalent to providing in this sentence below, if or in case?
The deposit is refundable when you leave, providing you haven't damaged anything.
In OP's context, providing is equivalent to if (it can validly be replaced by if, but not by in case).
If some statement S depends on the hypothetical scenario X being true (or false, as in not X or !X)...
S [is true/will happen] if X
S providing X
S so long as X
S as long as X
S unless (!X)
Note that OP's "hypothetical" is already negated (you have not damaged anything). If we remove the negation, it becomes !X, so an equivalent to OP's statement would be...
The deposit is refundable when you leave, unless you have damaged anything.
If some action A is only being taken to avoid the possibility of X being true (or to deal with issues that will only arise if X is in fact true)...
A lest X
A in case X
For example,...
Don't damage anything, lest you lose your deposit.
We ask for a deposit in case you damage anything.
In this sentence, the word providing means "on the condition." In other words, if you haven't damaged anything at the end of the lease, you will receive your security deposit back.
In this context, I would use the word assuming.
"The deposit is refundable, assuming that you have not damaged anything.