How to get rid of & in cases environment but still get the same space in the surroundings
You could use aligned
\documentclass{article}
\usepackage{amsmath}
\newenvironment{system}
{\[\left\lbrace\aligned}
{\endaligned\right.\]\ignorespacesafterend}
\begin{document}
Example:
\begin{system}
&x+y=2\\&x=2
\end{system}
\end{document}
Remove the intercolumn space and also the null delimiter space.
\documentclass{article}
\usepackage{amsmath}
\makeatletter
\newenvironment{system}
{%
\let\@ifnextchar\new@ifnextchar
\left\lbrace
\def\arraystretch{1.2}%
\array{@{}l@{}}%
}
{%
\endarray
\right.\kern-\nulldelimiterspace
}
\makeatother
\begin{document}
\[
\begin{system}
x+y=2,\\x=2
\end{system}
\equiv
\begin{system}
2+y=2,\\x=2
\end{system}
\]
\end{document}
I have doubts your readers will understand what you mean here.
\documentclass{article}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage[usestackEOL]{stackengine}
\stackMath
\setstackgap{L}{16pt}
\begin{document}
\[
\biggl\{\Centerstack[l]{x+y=2,\\x=2}
\equiv
\biggl\{\Centerstack[l]{2+y=2,\\x=2}
\]
\end{document}