Noticed something weird in derivation of relationship between ICBO and ICEO in BJTs

Question

So I'm currently studying Electronic Devices and Circuit Theory by Boylestad and Nashelsky (11th ed.) and I'm confused as to how the relation between \$I_{CEO}\$ and \$I_{CBO}\$ is derived. So the derivation in the book starts with the equation $$I_C = \alpha I_E + I_{CBO}\ \tag1$$ But since \$I_E = I_C + I_B\$ , $$I_C = \alpha (I_C + I_B) + I_{CBO}\ \tag2$$ Rearranging yields $$I_C = \frac{\alpha I_B}{1-\alpha} + \frac{I_{CBO}}{1-\alpha}\ \tag3$$ Then it goes on to say that \$I_C\$ when \$I_B = 0\$ is assigned the notation \$I_{CEO} = \frac{I_{CBO}}{1-\alpha} \approx \beta I_{CBO}\$ which is consistent with other sources.

The weird thing I noticed is that letting \$\alpha I_E = I_C\$ in eq. 1 produces $$I_C = I_C + I_{CBO}\ \tag4$$ and letting \$ \frac{\alpha I_B}{1-\alpha}=\beta I_B = I_C\$ in eq. 3 produces $$I_C = I_C + \frac{I_{CBO}}{1-\alpha}\ \tag5$$ which are surprisingly not the same in addition to being paradoxy/fallacious... So my question is what went wrong?

Answer 1

The original equations are: $$I_C = \alpha I_E + I_{CBO} \tag{a}$$ $$I_C = \beta I_B + (\beta +1)I_{CBO} \tag{b}$$

Before equation 4, you mentioned that \$\alpha I_E = I_C\$. But this is valid only when \$I_{CBO} = 0\$ and that is what equation 4 says.

In the same way, \$I_C = \beta I_B\$ is valid only if \$I_{CBO} = 0\$ and that is what equation 5 says.

Both equation 4 and 5 are valid only under the approximation that \$I_{CBO} = 0\$.