I have been given the axiom of irrelevance called "weak union" in the form $ X\perp Y,W|Z\implies X\perp W|Z,Y $. I understand how to interpret this, however the given proof is very confusing. Verbatim: $ X\perp Y,W|Z \implies p(x,y,w|z)=p(x|z)p(y,w|z) \\ \implies p(y|z)p(x,w|z,y)=p(x|z)p(y|z)p(w|y,z) \\ \implies p(x,w|z,y)=p(x,z)p(w|y,z) \\ \implies p(x,w|z,y)=p(x|z,y)p(w|y,z)\implies X\perp W|Z,Y $
The last line used the decomposition axiom. Is this not overkill? Why can I not go immediately to the last line, invoke the decomposition axiom and Bayes theorem to get the result immediately? I cannot see why lines 1-3 are needed, and what role they play in the final line. The jump from line 2-3 is self explanatory but 2 doesn't seem necessary, can we not start at $p(x,w|z,y)$?
Any help is appreciated.