What does an equal sign mean in a parenthesis?

Question

I'm now learning about Reinforcement Learning and in the Bellman equation, there is a following equal sign.

I understand the meaning of the left side of the equation, which is the probability of transitioning from state s to state s' with action a. However on the right side, what does the = sign mean?

The pr likely means the probability (but then why does it not use P?), and the | means given the condition of the right side, right? But I don't understand what the equal sign here means, and why is needed... IOW, what makes it different from pr(s_t+1 | s_t, a_t)?

Answer 1

The right hand side would be read:

"The conditional probability that $s_{t+1}$ is equal to $s'$ given that the value of $s_t$ is equal to $s$ and that the value of $a_t$ is equal to $a$."

Answer 2

It means the probability that the state at time $t+1$ is $s'$ given that the state and action at time $t$ are $s$ and $a$.