Trying to simplify the following boolean expression

Question

Simplify:

$y \times [x + (x' \times y)]$

My attempt:

1. $y \times [(x + x') \times (x + y)] \quad \textit{First Distributive Axiom} $
2. $y \times [1 \times (x + y)] \quad \textit{First Inverse Axiom} $

And for the continued steps, I am unfortunately stuck again.

Axioms available:

Answer 1

I don't see $yy=y$ directly in the axioms you have, but it follows from $yy=y(y+0)=y$. And so

$$ y(x + x'y)=yx+yx'y=yx+yyx'=yx+yx'=y(x+x')=y(1)=y. $$

Answer 2

Solution:

1. $y \times [(x + x') \times (x + y)] \quad \textit{First Distributive Axiom} $
2. $y \times [1 \times (x + y)] \quad \textit{First Inverse Axiom} $
3. $y(x+y) \quad \textit{Identity Axiom} $
4. $yx + yy \quad \textit{Distributive Axiom} $
5. $(yx) + y \quad \textit{Idempotent Axiom} $
6. $y + (yx) \quad \textit{Commutative Axiom} $
7. $y \quad \textit{Absorption Axiom} $