The opposite of a trivial truth is false; the opposite of a great truth is also true.History (or at any rate a brief google), does not record whether Bohr considered the above statement itself to be a great truth.
There is, however, another phrasing The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth. which, whilst it may initially appear to be less paradoxical than the first version, is in my opinion actually more so (or at the very least, more interestingly so). If you disagree, then tell me - what is the opposite of 'may well be'?
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It's all word-games of course, and I think the point is simply 'when you come across an assertion which appears to be greatly meaningful or resonant, it is often enlightening to consider its opposite as well'.