Suppose we have a perceptron with input neurons $x$ and weights $w$ and a threshold $\epsilon \in \mathbb{R}$
The activation function is the step function
$$a_{\epsilon}= \text{if} \ x > \epsilon \ \text{then} \ \text{else} \ 0$$
To activate the perceptron
$$\sum_{i=1}^nx_iw_i > \epsilon$$
Now when we have two perceptrons $p_1 =(x,w,\epsilon_1)$ and $p_2=(x,\tilde w,\epsilon_2)$ with $\epsilon_1,\epsilon_2 \in \mathbb{R}$
I want to show that $p_2$ is activated iff $p_1$ is activated $\forall\epsilon_1,\epsilon_2$
$$p_1 =(x,w,\delta_1)=\sum_{i=1}^nx_iw_i > \epsilon_1$$
$$p_2 =(x,w,\delta_1)=\sum_{i=1}^nx_i\tilde w_i > \epsilon_2$$
But I don't know how the show the equivalence.
I would say they can only be equivalent if all the weights and threshold are the same, because you can consider the edge cases where all $x_i=0$ (and thus you need $\epsilon_1= \epsilon_2$), and all cases where $x_i = 1$ and $x_j=0$ for all $0 \le i \le n$ and $j \not = i$ (from which it will follow that $w_i = \tilde w_i$)
Just conceptually: the perceptron draws a $n$-dimensional hyperplane through the $n+1$-dimensional feature space, and for the two perceptrons to be equivalent, those hyperplanes must be exactly the same.