This is Cerf's theorem. Every orientation-preserving diffeomorphism of the 3-sphere is isotopic to the identity. 

J.Cerf, Sur les difféomorphismes de la sphère de dimension trois (Γ4=0), Lecture Notes in Mathematics, No. 53. Springer-Verlag, Berlin-New York 1968.

Although Hatcher proved more (and you can in principle use his techniques to prove Cerf's result), Cerf's argument gives a very nice general technique.  It is the birthpoint of "Cerf Theory" meaning studying 1-parameter families of smooth functions, showing they can be assumed to be Morse at all but finitely-many times, and describing the cubic singularities where the family fails to be Morse.

Well, technically there is the additional step that homeomorphisms have unique smoothings.  This goes back to Moise. 

Moise, Edwin E. (1952), Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Annals of Mathematics. Second Series 56: 96–114,

What you're observing is that these Cerf and Moise results were considered "common knowledge" back then, much like how people do not bother to cite basic results about the fundamental group in papers nowadays.  What is worthy of citation and what is not is very much time and context sensitive.