# Common Emitter small signal model for high frequencies simplification

While trying to calculate the high frequency pole frequency corner of the common emitter amplifier I came across the following small signal model (found here in page 332)

The small signal equivalent simplified by Miller effect usage shown seems to apply Miller's theorem to $$\C_\pi\$$ and $$\r_\pi\$$ using only the equivalent inputs impedance formula:

$$C_\pi' = (1 - v_e/v_b)\cdot C_\pi\quad r_\pi' = (1 - v_e/v_b)\cdot r_\pi$$

Where, curiously $$\r_\pi' = r_\pi + (b+1)\cdot R_4\$$. I do not understand how this is valid, or if the theorem is even being applied here. I am aware that the current source may be connected to the ground, since it does not alter the value fed to the output resistance. However, that leaves us with $$\C_\pi\$$ parallel to $$\r_\pi\$$ in series with $$\R_4\$$:

How can both impedances be equivalent and how do I reach the given coefficient $$\1 - v_e/v_b\$$ (where $$\v_e/v_b = \frac{g_m + g_\pi}{g_m + g_\pi + G_4}\$$, which is the gain calculated for the CC amplifier).