This is called De Morgan's law and is usually stated

NOT (a AND b) = (NOT a) OR (NOT b)

NOT (a OR b) = (NOT a) AND (NOT b)

You can research its algebraic proof on the internet.

Here's a truth table proof

a b ~(ab) ~a ~b ~a+~b
0 0 1 1 1 1
0 1 1 1 0 1
1 0 1 0 1 1
1 1 0 0 0 0

--Mark