Impulse response (general form for linear systems as two-variables function $h(t,\tau)$) applied to Time-invariant systems
If the response to $x(t)$ is given by

$$y(t)=\int_{-\infty}^{\infty}h(t,\tau)x(\tau)d\tau\tag{1}$$

then the response to $x(t-T)$ is

$$\tilde{y}_T(t)=\int_{-\infty}^{\infty}h(t,\tau)x(\tau-T)d\tau=\int_{-\infty}^{\infty}h(t,\tau+T)x(\tau)d\tau\tag{2}$$

If the system is time-invariant we require

$$\tilde{y}_T(t)\stackrel{!}{=}y(t-T)=\int_{-\infty}^{\infty}h(t-T,\tau)x(\tau)d\tau\tag{3}$$

For $(2)$ and $(3)$ to be equal for any $x(t)$ we must have

$$h(t,\tau+T)\stackrel{!}{=}h(t-T,\tau)\tag{4}$$

which is equivalent to

$$h(t+T,\tau+T)\stackrel{!}{=}h(t,\tau)\tag{5}$$

From $(5)$ it is clear that for a linear and time-invariant system the value of $h(t,\tau)$ only depends on the difference of its arguments, and not on their individual values. Consequently, the impulse response can be rewritten as a function of $t-\tau$:

$$h(t,\tau)=\tilde{h}(t-\tau)\tag{5}$$