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Let's two events $S1$ and $S2$ are conditionally independent given the event $A$, i.e.,

$P(S_1|S_2,A) = P(S_1|A)$ and $P(S_2|S_1,A) = P(S_2|A)$

If $B$ is an arbitrary event, does the following probability hold?

$P(S_1|S_2,A,B) = P(S_1|A,B)$?

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No. We can ignore $A$-let it be certainly true. Let $S_1$ be a coin flip is heads and $S_2$ be a die roll is even. Now let $B$ be coin flip heads iff die roll is even. Then $P(S_1|S_2,B)=1, P(S_1|B)=\frac 12$

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