Why are the systems equivalent?
Upsampling (by $M$) and downsampling (by $N$) are interchangeable if and only if $M$ and $N$ are co-prime (i.e., they have no common factors other than $1$). This is the case in your example because $2$ and $3$ are co-prime.

Think about your specific example in the time domain. Introduce a signal $u[n]$ between the upsampler and the downsampler. In the first case you have

$$u[n]=\begin{cases}x[n/3],&amp; n=3k,\quad k\in\mathbb{Z}\\0,&amp;\text{otherwise}\end{cases}$$

and

$$y[n]=u[4n]=\begin{cases}x[4n/3],&amp; 4n=3k,\quad k\in\mathbb{Z}\\0,&amp;\text{otherwise}\end{cases}\tag{1}$$

In the second case you have

$$u[n]=x[4n]\tag{2}$$

and

$$y[n]=\begin{cases}u[n/3],&amp; n=3k,\quad k\in\mathbb{Z}\\0,&amp;\text{otherwise}\end{cases}\tag{3}$$

With $(2)$ you get

$$y[n]=\begin{cases}x[4n/3],&amp; 4n=3k,\quad k\in\mathbb{Z}\\0,&amp;\text{otherwise}\end{cases}\tag{4}$$

which is identical with $(1)$.