# Central Limit Theorem formula transformation with iid variables

I was looking into Central Limit Theorems and how a CLT is derived and I found this source quite helpful. The only thing I am having trouble to comprehend is the transformation of the formula

$$\frac{\overline X_n - E[\overline X_n]}{\sqrt{Var[\overline X_n]}}\overset{d}{\rightarrow}Z$$

in the case of iid variables. With iid variables, the formula becomes

$$\sqrt{n}\frac{\overline X_n - E[X_i]}{\sqrt{Var[X_i]}}\overset{d}{\rightarrow}Z$$

because of

$$E[\overline X_n]=E[X_i]\quad \text{ and }\quad Var[\overline X_n]= \frac{Var[X_i]}{n}.$$ How can the two assumptions regarding the expected value and the variance be derived fromm iid variables and why is the variance of a single observation i divided by the number of observations n the same as varaince of the sample mean?

If you have IID random variables $$X_1, X_2, ..., X_n$$, then the average $$\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$$ has expectation $$E(\bar{X}_n) = \frac{1}{n}\sum_{i=1}^n E(X_i) = E(X_1)$$. This previous fact has nothing to do with the fact that we have independence though! This is just from the identical distribution of the $$X_i$$'s and linearity of expectation.
We use the independence assumption when we calculate $$Var(\bar{X}_n) = Var \left(\frac{1}{n}\sum_{i=1}^n X_i\right) = \frac{1}{n^2}\sum_{i=1}^n Var(X_i) = \frac{1}{n^2} \cdot n Var(X_i) = \frac{Var(X_1)}{n}.$$
Notice in both cases the $$X_1$$ in the final expressions can be replaced by any of the $$X_i$$'s, since they are all identically distributed.