Centre justifying equations in align environment

Centre justifying equations in align environment

Apologies if this has been asked before, but I couldn't find it after
searching in a few places. I'm currently working with arrays of equations
like the following:
\begin{align}
y_i^3=y_i^2&=z_i^2=z_i^3 &&\text{if\, $1\leq i\leq n-1$}\\
y_iz_iy_i=y_i,\quad &z_iy_iz_i=z_i &&\text{if\, $1\leq i\leq n-1$}\\
y_iz_i&=z_{i+1}y_{i+1} &&\text{if\, $1\leq i\leq n-2$}
\end{align}
But I'd really like to drop the &'s in the equations themselves, so that
the equations are centre justified, the "if..." bits are left justified,
and I still have labels on each line. If I drop the &'s and change the
&&'s to &&&'s, then I get everything I want except that the equations are
right justified.
Can anyone help?