Why does this proof points out that $AB=AC$ and $AC=AB$?
He starts the proof remarking: "Let us conceive this triangle as two triangles". Try to think of the second triangle as $ A'B'C' $. With this change the two equalities become
$ AB = A'C' $ and $ AC = A'B' $.
AB=AC and AC=AB are not related to each other. This is just the "S","S" part of the SAS.