You don’t impeach someone who was never legitimately sworn in.
Case law, please? Or any legal opinions rendered in court? The Constitution provides only one way to remove a sitting President from office, and that is impeachment.
And in fact, we even have an example of fraudulently attaining an elected position in Robert Wexler (living outside his district). He's still the Congressman for a district where he doesn't reside, clearly against the Constitution, but it will take Congress to bump him out.
Essentially, the body charged with throwing out Congressmen, SC Justices, or the President or Vice President is Congress. If they do not take it up, then he's in the office.
You dont impeach someone who was never legitimately sworn in.
He was legitimately sworn in under potentially false premises. Impeachment is the remedy, per the Constitution and high crimes and misdemeanors which includes perjury. Perjury is violating an oath, which Obama could not have done until he was sworn in to office.