Why did he multiplied with the emitter current?
For amplifiers there are design equations and analysis equations.  The design equations are slightly simplified, in order to make the process of picking parts and currents easier.
Then once you have a first order design done, you go back and use the analysis equations to make sure you have met all the design goals and tweak as needed.
For first order design equations, Ie = Ic.
For analysis equations Ie = Ic + Ib.