Inside an event B of positive probability, probabilities are divided by P(B). Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.
Statement
A probability space \((\Omega,\mathcal F,\mathbf P)\) separates outcomes, measurable events, and their probabilities. A random variable \(X:\Omega\to\mathbf R\) must be measurable.
The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.
Worked algebra
A worked instance is useful here because it exposes every index that the compressed statement hides.
Conceptual compression
The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.
Caveat
Conditioning on a probability-zero event cannot be done by naïvely dividing. Conditional densities and regular conditional probabilities require additional structure.
The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.