# Side brace only some equations

With stmaryrd, nicematrix and tikz is relative easy:

\documentclass{article}
\usepackage{stmaryrd}
\usepackage{nicematrix}
\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing,
calligraphy}
\tikzset{
B/.style = {decorate,
decoration={calligraphic brace, amplitude=4pt,
raise=1mm},
very thick,
pen colour=red}
}

\begin{document}
$\begin{NiceArray}{c @{\,} c<{\strut}}% [create-extra-nodes, code-after = {\begin{tikzpicture} \draw[B] (1-2.north east) -- node[right=2mm] {four equation} (4-2.south east); \end{tikzpicture} } ] \llbracket \varphi \rrbracket_v & = \{\overline{(w, \sigma)} \in N_V \mid (w, \sigma) \models \varphi \} \\ \llbracket \varphi \rrbracket_v \llbracket \varphi \rrbracket_v & = \{\overline{(w, \sigma)} \in N_V \mid (w, \sigma) \models \varphi \} \\ \llbracket \varphi \rrbracket_v & = \{\overline{(w, \sigma)} \in N_V \mid (w, \sigma) \models \varphi \} \\ \llbracket \varphi \rrbracket_v & = \{\overline{(w, \sigma)} \in N_V \mid (w, \sigma) \models \varphi \} \\ \llbracket \varphi \rrbracket_v & = \{\overline{(w, \sigma)} \in N_V \mid (w, \sigma) \models \varphi \} \end{NiceArray}$
\end{document}


You need to compile at least twice that you will get showed result.

A simple solution using an array and the bigdelim package:

\documentclass{article}
\usepackage{array, bigdelim}
\usepackage{stmaryrd}
\usepackage{mathtools}

\begin{document}

\begin{equation*}
\begin{array}{r@{}l@{\:}l}
\llbracket\varphi\rrbracket_v &{}=\bigl \{\overline{(w, \sigma)} \in N_V \mid (w, \sigma) \models \varphi\bigr\}&\rdelim\}{4}{*}[\text{\quad three equations}] \\[1ex]
\llbracket\varphi\rrbracket_v &{}=\bigl \{\overline{(w, \sigma)} \in N_V \mid (w, \sigma) \models \varphi \bigr\} \\[1ex]
\llbracket \varphi \rrbracket_v &{}=\bigl \{\overline{(w, \sigma)} \in N_V \mid (w, \sigma) \models \varphi \bigr\} \\[1ex]
\llbracket \varphi \rrbracket_v &{}=\bigl \{\overline{(w, \sigma)} \in N_V \mid (w, \sigma) \models \varphi \bigr\}
\end{array}
\end{equation*}

\end{document}