Is this correct reasoning?
Let $x_i$ be a variable in a Bayesian Network and $\text{MB}(x_i)$ denotes its Markov blanket.
Let us note that: $$ p(x_i \mid \text{MB}(x_i)) \propto p(x_i, \text{MB}(x_i)). $$
By the definition of Bayesian network (factorisation property): \begin{multline} p(x_i, \text{MB}(x_i)) = p(x_i \mid \text{Parents}(x_i)) \ \times \\\prod_{y_j \in \text{Children}(x_i)} p(y_j \mid \text{Parents}(y_j)) \prod_{z_k \in \text{Co-parents}(x_i)} p(z_k \mid \text{MB}(z_k)). \end{multline}
Therefore: $$ p(x_i \mid \text{MB}(x_i)) \propto p(x_i \mid \text{Parents}(x_i)) \prod_{y_j \in \text{Children}(x_i)} p(y_j \mid \text{Parents}(y_j)), $$
since $ \prod_{z_k \in \text{CoParents}(x_i)} p(z_k \mid \text{MB}(z_k))$ becomes constant (it does not depend on $x_i$, the value of od variable $z$ is fixed and $p(z_k \mid \text{MB}(z_k))$ becomes constant).