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Conditional independence $A\perp B\mid D$ means that, for almost every $d$,equivalently $p(a\mid b,d)=p(a\mid d)$ wherever the conditional probabilities are defined.

Now use the chain rule and both assumed independences:This is exactly $A\perp (B,C)\mid D$, proving the contraction axiom for conditional independence.

Let $M$ be a Markov blanket of the treatment $A$ inside $(A,X)$, and write the remaining adjustment variables as $N=X\setminus M$. By definition,The sufficiency of $X=(M,N)$ givesApply the contraction axiom for conditional independence with first variable $A$, second variable $N$, third variable $Y(a)$, and conditioning variable $M$. It givesThe decomposition axiom for conditional independence then yields $A\perp Y(a)\mid M$. Hence every Markov blanket of $A$ in $(A,X)$ is itself a sufficient adjustment set.