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}