Obama used a Bible at the official inaugural on January 20th, but the Chief Justice messed up the words. Obama repeated the messed up words. So that oath was not valid. It did not have the Constitutionally required text.
Obama retook the oath on January 21st, with the correct words, but no Bible. That oath was valid because it matched the text in the Constitution.
It was valid either way (you think any judge would take the case of the screwed up oath, knowing that it’d embarrass the Chief Justice for making an innocent mistake). Chief Justice Roberts felt so bad about it though (it was his first of what I’m sure will be many inaugurations), he wanted a do over and the President agreed.
Besides, there’s an old legal maxim— equity regards as done that which ought to have been done. If the Chief Justice meant to give the right words and the President meant to repeat the right words... then that’s good enough for government work (to quote an old NASA saying).
Finally, George Washington set the precedent that the President takes power the moment his term begin, GW didn’t take the oath until more than a month after he was on the job.
http://www.law.cornell.edu/anncon/html/art2frag6_user.html
PLS.: witnesses? video?