If we knew that in an unfair voting system (ie: it is known that IIA does not hold) then if (1) holds, how would we go about inferring that (2) holds and then is it possible to show that if (1) and (2) hold , it implies (3)?
1) There is no "dictator": no single voter possesses the power to always determine the group's preference.
2) If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
3)If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
Thanks
(1) does not imply (2), and (1) and (2) together do not imply (3). There are many non-dictatorial methods that fail to satisfy (2); see this Wikipedia article. For a simple method that satisfies (1) and (2) but not three, let $u$ and $v$ be two designated voters. If $u$ and $v$ have the same preferences, their preferences are adopted by society; if not, range voting is used. There is no dictator, and (2) is satisfied, but (3) clearly is not.