I have problems understanding constraints on correlation functions of quasi primary fields (QPF) following DiFrancesco's Conformal field theory book. In chapter 4, section 4.2.1, a QFP is defined as a field with the following transformation law under conformal transformations
$$ \phi(x) \rightarrow \phi' (x') = \Biggl\vert{\frac{\partial x'}{\partial x}} \Biggl\vert ^{-\Delta/d} \phi(x) \tag{1} $$
In section 4.3.1 constraints on 2-point correlation functions are found, we have: $$ \langle \phi_1(x_1) \, \phi_2(x_2) \rangle = \Biggl\vert{\frac{\partial x'}{\partial x}} \Biggl\vert ^{-\Delta_1/d}_{x=x_1} \, \, \Biggl\vert{\frac{\partial x'}{\partial x}} \Biggl\vert ^{-\Delta_2/d}_{x=x_2} \langle \phi_1(x'_1) \, \phi_2(x'_2) \rangle \tag{2} $$
Specializing to a scale transformation $x'=\lambda x$ we have
$$ \langle \phi_1(x_1) \, \phi_2(x_2) \rangle \, = \,\lambda^{\Delta_1 + \Delta_2} \, \langle \phi_1(\lambda x_1) \, \phi_2(\lambda x_2) \rangle \tag{3} $$
Rotations and translations invariance require $$ \langle \phi_1(x_1) \, \phi_2(x_2) \rangle \, = \, f(\mid x_1 -x_2 \mid) \tag{4} $$
Where $$ f(x)\, = \, \lambda^{\Delta_1 + \Delta_2} \, f(\lambda x) \tag {5} $$
In other words $$ \langle \phi_1(x_1) \, \phi_2(x_2) \rangle \, = \, \frac{C_{12}}{\mid x_1-x_2 \mid^{\Delta_1+ \Delta_2}} \tag{6} $$
This passage really confuses me, why isn't it just
$$ \langle \phi_1(x_1) \, \phi_2(x_2) \rangle \, = \, \lambda^{\Delta_1+ \Delta_2} \, f(\lambda \mid x_1 -x_2 \mid) \tag{7} $$
As I think it should follow from $(5)$?
I don't get why it has that specific form showed in $(6)$ and not the general one showed in $(7)$.