What is the logical equivalence of 'provided that...'? Simple question as above.
Given "A provided that B",
the logical equivalence seems to be "A only if B". 
Very startling that my lecturer enjoy using such 'informal' speak in his lecture.
Confirmation please.
 A: It's not informal: just another way of saying "as long as", "when", or "if" in a narrative.
"$n$ is not prime provided that it is even and sufficiently large", for example, is equivalent to "if $n$ is even and sufficiently large, then it is not prime".
If you find "provided that" informal, why do you accept "if"? They're all unambiguous. You could ask your professor to be more limited in their vocabulary, but it's the kind of thing it's very hard to notice yourself doing. It is good practice, though, when stating theorems formally, to use "if… then…", just because it's easiest to understand at a glance.
A: It is equivalent to "if." By the way, one could claim that your use of "only if" is just as 'informal' as your professors use of 'provided that.' "Only if" could potentially be confused with "if and only if," which DOES have a different meaning.
I hope this doesn't come off as condescending at all, but I think there is a phenomenon sometimes of people first learning how to "speak math" and then using it very strictly, only to learn that, as a practical manner, we often use language which you might call 'informal.' As someone else pointed out, 'if' itself doesn't really have a definition. At some point we must use these words and move on.
Lastly, if you happen to use english as a second language, then that could be an alternative source of confusion, which is a different discussion.
