A quandle is a nonempty set $X$ on which there is defined a binary operation $(x,y) \rightarrow x*y$ satisfying the following properties.
(Q1) $x*x=x$, $\forall x \in X$.
(Q2) For any $x,y \in X$, there is a unique element $z \in X$ such that $x=z*y$.
(Q3) For any $x,y,z \in X, x*(y*z)=(x*y)*(x*z)$
Let $X$ be a finite quandle, and $G$ be an abelian group written multiplicatively. A map $\theta:X \times X \times X \rightarrow G$ is called a $3$-cocycle of $X$ with the coefficient group $G$ if it satisfies the following two conditions:
(i) $\theta(x,x,y)=\theta(x,y,y)=1_G$, for any $x,y \in X$.
(ii) For any $x,y,z,w \in X$
$\theta(x,z,w)-\theta(x,y,w)+\theta(x,y,z)=\theta(x*y,z,w)-\theta(x*z,y*z,w)+\theta(x*w,y*w,z*w)$
My question is the following: Suppose $x*z_1=x*z_2=x$ and $y*z_1=y*z_2=y$, where $x,y,z_1,z_2$ are distinct elements in the quandle $X$. Then is the following equation true
$\theta(x,y,z_1)=\theta(x,y,z_2)$?
Any help is highly appreciated