Equivalent circuit and substitution
I don't know what a "rigorous proof" would be, but consider that you combined R2 and R3 because they are parallel. By definition, elements in parallel have the same voltage across them, and as you observe the equivalent resistor must also have the same voltage across it. So, if you find the voltage across R23 then you have also found the voltage across R2 and R3. This is because of how the terms parallel and equivalent are defined in this context.

Based on OP's edits: When you combine elements in parallel the voltage across them is the same by definition, and is the same across the equivalent resistor that replaces them, but the current through them may be different. When you combine elements in series the voltage across them may be different but the current through them is the same by definition, and is the same through the equivalent resistor that replaces them.

This is inherent in the definition of parallel and series. I don't know how they would be proved more rigorously.