When learning about BCFW recursion I am shown the deformation equation:

$$\hat{p_1}=p_1 -zq \hspace{5mm}; \hspace{5mm} \hat{p_n} = p_n +zq$$

This deformation represents a $\langle1n]$ shift and $q$ is defined as $q=\lambda^\dot{\alpha}_1 \lambda^\alpha _n $

To check under which circumstances $\lim _{z \to \infty} \hat{A_n}(z)=0$, we consider deforming the 4-point MHV amplitude $A_4(1^- 2^- 3^+ 4^+)$.

Supposing we do a $\langle 1^- 2^-] $ shift then:

$$\hat{A_4}^{--}= \frac{\langle \hat{1}\hat{2}^3 \rangle}{\langle \hat{2}3 \rangle \langle 34\rangle \langle 4\hat{1} \rangle}$$

This is where my problem arises:

In the notes, I am told to note that:

$$\langle \hat{1} \hat{2}\rangle= \left( \langle1|-z\langle 2|\right)|2\rangle =\langle 12\rangle $$

where $\langle 22\rangle=0 $ has been used.

How is this last equation true? I don't understand how this last equality was worked out or how it makes $\langle \hat{1} \hat{2} \rangle = \langle 12 \rangle$ I am pretty sure the first two equations are used in it but I don't see how.