# Is this $\alpha$-conversion correct?

I found this $$\alpha$$-conversion below in a certain book:

$$(\lambda x. x(\lambda z. xy)) = \alpha (\lambda z. z(\lambda x. zy))$$

Well... this seems wrong to me, since $$z$$ is a binding variable in $$(\lambda x. x(\lambda z. xy))$$.

Am I just not understanding?

• Three comments on proper use of this site: (1) Please use MathJax to format all mathematical expressions. I have edited your post to add proper MathJax formatting. You can press edit to see the code for how I did this. In particular, $\alpha$ and $\lambda$ are \alpha and \lambda inside dollar signs. (2) You say you saw something in a "certain book". What book? Always cite your sources! (3) No need to write "I ask for help." This is understood! Oct 1 at 19:51
• But they changed the variable to $x$ in the second expression. You can think of this as happening over a couple steps, all $\alpha$-conversions: $$\lambda x.x(\lambda z.xy)\to \lambda x.x(\lambda u.xy)\to \lambda z.z(\lambda u.zy)\to \lambda z.z(\lambda x.zy)$$ Oct 1 at 20:07
• Got it! Thank you very much! Oct 1 at 21:45