Pagebreaks in an aligned environment.

I can get the results that I want, but the syntax does not make sense
to me.  So I thought I'd try to augment my understanding.

I found from web-searching that a fundamental requirement is to put

    \allowdisplaybreaks

in the preamble.  This has the desired effect, but my situation is
that I am not happy with the point at which LaTeX chooses to put the
break.  I tried putting in a \pagebreak command at the point where I
wanted the break, but this command seemed to be ignored.

More web-searching led me to the \displaybreak command. This "works",
but what puzzles me is that I have to put this command one line
*before* the point at which I want the break to come.  The attached
file demo.tex demonstrates the phenomenon.  In order to get the break
to come after the third displayed equation, I have to put the
\displaybreak command after the *second* equation.  This makes no sense
to me.

Can anyone explain to me what I am missing?  Or do I just have to say
"Them's the breaks"? 🙂️

cheers,

Rolf

-- 
Honorary Research Fellow
Department of Statistics
University of Auckland
Stats. Dep't. (secretaries) phone:
         +64-9-373-7599 ext. 89622
Home phone: +64-9-480-4619

\documentclass{article}
\usepackage{amsmath}
\allowdisplaybreaks
\begin{document}
\begin{align*}
x &= 1 \\
y &= 2 \\ \pagebreak
z &= 3 \\
a &= 4 \\
b &= 5 \\
c &= 6 \\
\end{align*}
\end{document}