Context free languages invariant by "shuffling" right hand side
Your theorem doesn't hold.
Consider the grammar
$$
\begin{align}
&amp;S \to 1T \mid T1 \\
&amp;T \to 23 \mid 32 \\
\end{align}
$$
Shuffling the grammar results in the same language.
Now consider instead the grammar
$$
S \to 123 \mid 132 \mid 231 \mid 321
$$
which generates the same language.
Shuffling this grammar results in a larger language.