Reason for certain steps in Circuit Analysis
Your professors analysis and lecture is very good.  Your question above is out of context and as such you've really kneecapped anyone who is going to analyze it without them understanding that this is really just part of an analysis and rationale for how you do small signal analysis.  He is taking you through replacing the non-linear elements and substituting in the small signal equivalents (all lower case letters as is the convention).

I would recommend going back to ~ 18 minute mark and listening carefully.

He has removed the large signal currents (capital letters above - as is the convention) because they represent the operating point (or network bias).  When you are doing small signal analysis you don't consider those currents you only care about the incremental (small signal) currents and voltages.  His approach is a more rigorous (i.e. proper) derivation.