electromagnetism - Obtain the same Maxwell's equation after a change of coordinates

In the usual $(x,y,z)$ system of coordinates, if we expand the Maxwell's curls equations for phasors

$$\nabla \times \mathbf{E} = - \mathbf{J}_m - j \omega \mu \mathbf{H}$$ $$\nabla \times \mathbf{H} = \mathbf{J} + j \omega \epsilon \mathbf{E}$$

we obtain equations like the following:

$$\displaystyle \frac{\partial H_y}{\partial z} - \frac{\partial H_z}{\partial y} = J_{x} + j \omega \epsilon E_x\tag{1}$$

Now, let $\mathbf{E}'$, $\mathbf{H}'$, $\mathbf{J}'$, $\mathbf{J}_m'$ be a new field (with its new sources). In particular we have

$$H'_y = H_y$$ $$H'_z = - H_z$$ $$J'_x = - J_x$$ $$E'_x = - E_x$$

The equation (1) for this field becomes

$$\displaystyle \frac{\partial H'_y}{\partial z} - \frac{\partial H'_z}{\partial y} = J'_{x} + j \omega \epsilon E'_x\tag{2}$$

If we define $x' = x$, $y' = y$, $z' = -z$, (2) becomes

$$\displaystyle -\frac{\partial H'_y}{\partial z'} - \frac{\partial H'_z}{\partial y'} = J'_{x} + j \omega \epsilon E'_x$$

that is, substituting the primed quantities with the equivalent unprimed ones,

$$\displaystyle - \frac{\partial H_y}{\partial z'} + \frac{\partial H_z}{\partial y'} = -J_{x} - j \omega \epsilon E_x$$

$$\displaystyle \frac{\partial H_y}{\partial z'} - \frac{\partial H_z}{\partial y'} = J_{x} + j \omega \epsilon E_x\tag{3}$$

The equation (2) can be written in the form (3): so, we have obtained the same equation as (1), but with respect to the primed variables!

Questions: can (3) still be considered a Maxwell's equation as well as (1)? Why?

My question arises because (3) is weird: it involves components of fields along the unprimed unit vectors ($\mathbf{u}_x$, $\mathbf{u}_y$ and $\mathbf{u}_z$), and derivation with respect to the primed variables ($x'$, $y'$ and $z'$).

In particular, $H_z$ has a different behaviour according to the system of coordinates, because $\mathbf{u}_z = - \mathbf{u}_{z'}$. So, how can this be taken into account while evaluating (3)? This is what I can't understand.

This demonstration is used to derive the fields in the Method of the Images in the Electromagnetic theory (the primed field is the image field).