Align equal sign in a big block of equations

I would like to align the equal signs in this block of equations. Somehow, the first equal sign is well aligned, but the second is not. Am I missing something obvious?

Thank you very much

\documentclass{article}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{amsmath}
\begin{document}

\begin{equation}
\left\lbrace
\begin{aligned}
u_0 &=p_0 \\
u_1 &=p_0+p_1&=u_0+p_1  \\
u_2 &=p_0+p_1+p_2&=u_1+p_2  \\
\vdots \\
u_{n-1} &=p_0+p_1+\dots + p_{n-1} &= u_{n-2}+p_{n-1}  \\
u_{n} &=p_0+p_1+\dots + p_{n} &= u_{n-1}+p_{n}  \\
\end{aligned}
\right.
\end{equation}
\end{document}

Use alignedat. In contrast to aligned you have to specify the number of column pairs there, i.e. \begin{alignedat}{2}. Also ever third column is right aligned so you need to insert an empty column before your second equals as in &&=.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
\left\{
\begin{alignedat}{2}
u_0 &=p_0 \\
u_1 &=p_0+p_1&&=u_0+p_1  \\
u_2 &=p_0+p_1+p_2&&=u_1+p_2  \\
\vdots \\
u_{n-1} &=p_0+p_1+\dots + p_{n-1} &&= u_{n-2}+p_{n-1}  \\
u_{n} &=p_0+p_1+\dots + p_{n} &&= u_{n-1}+p_{n}  \\
\end{alignedat}
\right.
\end{equation}
\end{document} 