What chemical equation proves that KHCO3 can extinguish a fire?
According to this Wikipedia entry:


  Decomposition of the bicarbonate occurs between 100 and 120 °C (212
  and 248 °F):
  
  $$\ce{2 KHCO3 → K2CO3 + CO2 + H2O}, \Delta H&gt;0$$


So, you produce a nonflammable powder and two nonflammable gases, and you absorb heat from the fire in an endothermic reaction.  

The picture here is more complex than just a chemical equation however. The $\ce{KHCO3}$ also acts as an energy-absorbing material and provides a solid surface on which free radicals can be destroyed.