Full Adder boolean Algebra simplification

Question

I have an expression here from the Full Adder circuit, used for binary addition. One equation used to make it work, is this one:

$$C = xy + xz + yz \tag{1}$$

Now, the book transforms this equation into this: $$C = z(x'y + xy') + xy \tag{2}$$

In the immediate step, the do this: $$C = z(x \oplus y) + xy \tag{3}$$

Now, my question is how does the second equation come from the first one?

Answer 1

\begin{align*} C & = xy + xz + yz\\ & = xy + x(y + y')z + (x + x')yz \\ & = xy + xyz + xy'z + xyz + x'yz \\ & = xy(1 + z + z) + (xy' + x'y)z \\ & = xy + (xy' + x'y)z \\ & = xy + (x \oplus y)z \end{align*}