Proof of time-invariance of continuous-time system
Let $y_1(t)$ be the response to the signal $x_1(t)$:

$$y_1(t)=x_1(-t)\tag{1}$$

Now let $x_2(t)$ be a shifted version of $x_1(t)$:

$$x_2(t)=x_1(t-T)\tag{2}$$

The response to $x_2(t)$ is

$$y_2(t)=x_2(-t)=x_1(-t-T)\tag{3}$$

If the system were time-invariant, its response to $x_2(t)$ should be a shifted version of its response to $x_1(t)$:

$$y_2(t)=y_1(t-T)\tag{4}$$

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.