Halachic Logic Inquiry

Pinchas writes "If there is no Levi the Cohenim can wash their hands themselves or a firstborn can wash them."

I find this tidbit of information interesting, given my previous post linking a "line of kehunah" through the phenomenon of pidyon haben(ot).

Can anyone in the blogosphere elaborate on the halachic logic which establishes this "line of kedushah" linking the kohanim, levi'im and the firstborn?

technorati tags: Torah halachah talmud torah judaism kehunah jewish