For the standard binary field $\mathbb{F}_{2} = \{0, 1\}$. Where the operations of addition and multiplication exist, and multiplication is equivalent to logical and, and addition is equivalent to logical exclusive or.
The field axiom of distributivity of multiplication over addition is: $\forall x,y,z \in \mathbb{F}_{2} \mspace{8mu} x(y + z) = xy +xz $
I understand I can make a truth table to demonstrate this, and I have been demonstrating that the binary field fulfills other field axioms, but I was wondering if there was another way. My mathematical toolkit isn't very large, so perhaps this is limiting me from seeing another standard way to demonstrate this.