Linear programming: reduce a contstraint that includes minimun
Solve two linear programs.  The first has the constraints
$$
A \mathbf x = \mathbf b,\quad x_1 = x_2 \le x_3.
$$
The second has the constraints
$$
A \mathbf x = \mathbf b,\quad x_1 = x_3 \le x_2.
$$
Choose whichever solution gives you a smaller value for $\mathbf c' \mathbf x$.

This works for your particular situation.  It doesn't scale to a case where you have a large number of $\min$ constraints.