Let $y_1(t)$ be the response to the signal $x_1(t)$:
Now let $x_2(t)$ be a shifted version of $x_1(t)$:
The response to $x_2(t)$ is
If the system were time-invariant, its response to $x_2(t)$ should be a shifted version of its response to $x_1(t)$:
However, from (1) we have $y_1(t-T)=x_1(-(t-T))=x_1(-t+T)$. Comparing this to (3) we see that (4) is not satisfied, and, consequently, the system is not time-invariant. Now that you've seen the proof, try drawing all the signals in order to gain a better understanding.