The Torah explicitly tells us that when we wear a (four) cornered garment it needs tzitzis and Tcheles. ״ועשו להם ציצית על כנפי בגדיהם לדרותם ונתנו על ציצית הכנף פתיל תכלת״ (במדבר ט״ו:ל״ג)
Since today we are uncertain about what Tcheles is, how are we allowed to wear tzitzis?
Rambam mishnah Torah sefer ahava tzitzis ch. 1
Here the Rambam explains the mitvah tzitzis and techeles can be fulfilled independently of each other. If one does not have techeles then the white strings can be worn alone to fulfill that mitzvah. Similarly if one had both in their tzitzis but the white stings were somehow torn off so that only techeles remained you would still be permitted to wear them to fulfill the mitzvah of tzitzis.
Kli chemda says in mitzvas tzitzis is lidoros (fir all fenerations)and ticheles isnt ldoros (for generations, since it is destined to be lost)(based of drasha) therefore the aseh is more chashuv and it pushes off aseh of ( not wearing) ticheles. Rav perlow says similar, that mitzvas tzitzis is more chashuv than tichekes, so it pushes it off. But according to the Mordechai that holds if one cant put on strings he doesnt have to take off beged since he's ones(accident) the question doesn't start.
