I have the following equation: $$ P = y\bar{z} + \bar{x}(x + \bar{y}(y + \bar{z}(z + x))) + (x + z)(\bar{x} + y) + \overline{xy+\bar{x}\bar{z}+y\bar{z}} $$ When simplified it comes down to $x + y + z$.
I tried, but I can't seem to get to that solution. Can someone try to simplify it step by step? Thank you.
\begin{equation}\tag{1} P = y\bar{z} + \bar{x}(x + \bar{y}(y + \bar{z}(z + x))) + (x + z)(\bar{x} + y) + \overline{xy+\bar{x}\bar{z}+y\bar{z}} \end{equation}
To simplify $(1)$, it helps to break it up first. The second term can be expanded and evaluated in the way of $(2)$.
\begin{equation}\tag{2} \bar{x}(x + \bar{y}(y + \bar{z}(z + x)))\\ 0 + \bar{x}\bar{y}(y + \bar{z}(z + x)))\\ 0+ \bar{x}\bar{y}\bar{z}(z + x)\\ 0 \end{equation}
The third term in the way of $(3)$.
\begin{equation}\tag{3} (x+z)(\bar{x}+y)\\ 0+xy+z\bar{x}+zy \end{equation}
The forth term reduces to $(4)$, using De-Morgan's law.
\begin{equation}\tag{4} \overline{xy+\bar{x}\bar{z}+y\bar{z}}\\ (\bar{x}+\bar{y})(x+z)(\bar{y}+z)\\ (0+\bar{x}z+\bar{y}x+\bar{y}z)(\bar{y}+z)\\ \bar{x}z\bar{y}+\bar{x}z+\bar{y}x+\bar{y}xz+\bar{y}z+\bar{y}z\\ \bar{x}z+\bar{y}x+\bar{y}z\\ \end{equation}
Bringing it all together $(1~-~4)$, we simplify $P$.
\begin{align} P &= (y\bar{z})+0+(xy+z\bar{x}+zy)+(\bar{x}z+\bar{y}x+\bar{y}z)\\ &=(zy+z\bar{y})+(y\bar{z}+yz)+(xy+x\bar{y})\\ &=z+y+x \end{align}