Conditional probability with inequality

Question

It is known that: $P(X > a) = 1 - P(X \leq a)$
Is there a rule for $P(X > a | Y > b)$ ? Maybe something similar to:

$P(X > a | Y > b) = 1- P(X < a | Y < b)$ (I am not sure, just a guess)

Thank you!

Answer 1

$P(X > a | Y > b) = 1- P(X \le a | Y >b)$ is true since

\begin{align*} &P(X > a | Y > b)+P(X \le a | Y >b)\\ =&\frac{P( X>a \& Y>b)}{P(Y>b)}+\frac{P( X\le a \& Y>b)}{P(Y>b)}\\ =&\frac{P(Y>b)}{P(Y>b)}\\ =&1 \end{align*}

Answer 2

$P(X > a | Y > b) = 1- P(X \leq a | Y > b)$

\begin{align} &P(X > a | Y > b) = \frac{P(\{X>a\} \cap \{Y>b\})}{P(\{Y>b\})} = \frac {P(\{Y>b\})- P(\{X\leq a\} \cap \{Y>b\})}{P(\{Y>b\})} \\&= 1 - \frac{P(\{X\leq a\} \cap \{Y>b\})}{P(\{Y>b\}} = 1- P(X \leq a | Y > b) \end{align}