Is there a simple proof of this identity or a reference to some textbook where could I find a simple proof of the $(1)$?
$$\boldsymbol{\nabla}\times (\boldsymbol{\nabla}\times \mathbf{E})=-\frac{\partial }{\partial t}(\boldsymbol{\nabla}\times \mathbf{B})\tag{1}$$ where Maxwell’s version of Faraday’s law of induction is
$$\boldsymbol{\nabla}\times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$$
Deriving the equation
$\boldsymbol{\nabla}\times (\boldsymbol{\nabla}\times \mathbf{E})=-\dfrac{\partial }{\partial t}(\boldsymbol{\nabla}\times \mathbf{B})\tag{1}$
from
$\boldsymbol{\nabla} \times \mathbf E = -\dfrac{\partial{\mathbf B}}{\partial t} \tag 2$
is really more a matter of muliple variable calculus than electromagnetism. We are merely using the fact that, for sufficiently differentiable $\mathbf B$ (I'm pretty sure $C^2$ will do), the order in which we take partial derivatives is immaterial to the result. Therefore, taking $\boldsymbol{\nabla} \times$ of each side of (2) to obtain
$\boldsymbol{\nabla} \times (\boldsymbol{\nabla} \times \mathbf E) = -\boldsymbol{\nabla} \times \dfrac{\partial{\boldsymbol B}}{\partial t}, \tag 3$
we need simply interchange the order of the derivatives occurring on the right-hand side (that is, of $\partial / \partial t$ and $\boldsymbol{\nabla} \times$) and voila! (1) is obtained.