Using KVL in nodal analysis
Picture this:







simulate this circuit &ndash; Schematic created using CircuitLab

It looks pretty obvious that \$V_a = V_1 + V_2\$, but this is technically a form of KVL. If you take a loop clockwise, you've got:

\$V_a - V_1 - V_2 = 0\$

which is the same thing that your intuition tells you.

Essentially, whenever you make one of those substitutions, you're making a loop and applying KVL around it. The only weird thing is that not all of the pieces of your loop need to be components in the circuit - it's perfectly valid to make one of them a voltage jump, like this example.

EDIT:

You asked for another circuit. Here's my best shot:







simulate this circuit

If we go up the voltage supply and down the 'jump' in loop 1, we get
\$V_a - V_1 = 0\$
or \$V_1 = V_a\$.

If we 'jump' to V1, move across R1, and 'jump' from V2 to ground, we get
\$V_1 - V_{R1} - V_2 = 0\$
or \$V_{R1} = V_1 - V_2\$.

If we 'jump' to V2 and move back down R2, we get
\$V_2 - V_{R2} = 0\$
or \$V_{R2} = V_2\$.

Hopefully those three examples make sense.