Is there any equation that can be formed between $P(A | B)$ and $P(A| B')$, other than the Law of Total Probability?

Question

I am aware that by the Law of Total Probability.

$P(A) = P(A \cap B) + P(A \cap B')$

$P(B) = P(A | B) P(B) + P(A | B') P(B')$

But I was wondering if there was some sort of relationship between the probabilities $P(A | B)$ and $P(A | B')$ on their own, or without needing other variables. Maybe less variables if all the other ones could not be excluded.

I know this is not suppose to be related, but it is true that $P(A | B) + P(A' | B) = 1$

So I was just wondering if aside from the Law of Total Probability, there is another significant equation that can be formed if instead the occurred event became the complements and the event that will occur will remain constant.

Answer 1

I think the Law of Total Probability is the most fundamental one.

$$ P(A) = P(A|B)P(B) + P(A|B')P(B')$$

and it only involve the extra information $P(A), P(B)$