This is shown in 
MR1310882 (95m:05091) 
Negami, Seiya(J-YOKOED)
Diagonal flips in triangulations of surfaces. (English summary) 
Discrete Math. 135 (1994), no. 1-3, 225–232. 
You can [check it out.][1] You will note that the argument uses some heavy lifting, unlike Hatcher's.


  [1]: http://dl.dropbox.com/u/5188175/Flipsequiv.pdf