The power set $P(X)$ of a finite set $X$ is a bilinear space over $F_2$

Question

In the book of Kazimierz Szymiczek (Algebra, Logic and Applications) Bilinear Algebra An Introduction to the Algebraic Theory of Quadratic Forms, page 18. I have found that the power set $(P(X),+,\cdot)$ of a finite set $X$ is a bilinear space over the field $\mathbb{F}_2$ by the rules: $$\forall A,B\in P(X),\qquad A+B:=A\Delta B=(A\cup B)\setminus (A\cap B) $$ and the multiplication by a scalar is defined as $$0\cdot A =\emptyset,\qquad 1\cdot A =A.$$ Next he defined the bilinear form $f$ on $P(X)$ by: $$\forall A,B\in P(X),\quad f(A,B)=|A\cap B|\mod 2 $$ So for me, it is clear why $f(\lambda\cdot A,B)=\lambda\cdot f(A,B)$ for $\lambda\in \mathbb{F}_2$, but it is not clear why this form is bilinear, exactly my problem is how to prove the linearity with respect to the first argument: $$\big((A+B)\cap C\big)\ \text{ mod }2=(A\cap C)\ \text{mod } 2+(B\cap C)\ \text{mod 2}$$