Multivariate and integral representations explanation

I would to understand more on the multivariate and integral representations. I am fascinated by the possibility of a multivariate distribution like this one:

$$\mathbb{P}(X_1 \le Z + a, X_2 \le Z + a, ..., X_n \le Z + a),$$

to be transformed into:

$$\int \limits_{- \infty} ^{+ \infty} {\Phi(z+a)^n \phi(z)dz}.$$

and vice-versa. Here $$\Phi(z)$$ and $$\phi(z)$$ are, respectively, the cdf and the pdf. Everything started reading about this: Integrating pdf times cdf squared.

How is that a random vector is transformed into a scalar? is this some sort of projection? is there an analog transformation between a euclidean vector and a scalar?

Lets assume $$Z$$ is a random variable defined over a support of $$\mathbb{R}$$ and $$X_1,...,X_n$$ are i.i.d and have cdf $$\Phi$$, in which case
\begin{align} \text{Prob}(a) &\equiv \mathbb{P}(X_1 \le Z + a, X_2 \le Z + a, ..., X_n \le Z + a) \\[18pt] &= \mathbb{E}_Z[\mathbb{P}(X_1 \le Z + a, X_2 \le Z+ a, ..., X_n \le Z + a | Z = z)] \\[12pt] &= \int \limits_{-\infty}^{\infty} \mathbb{P}(X_1 \le Z + a, X_2 \le Z+ a, ..., X_n \le Z + a | Z = z) \phi(z) \,dz \\[6pt] &= \int \limits_{-\infty}^{\infty}\mathbb{P}(X_1 \le z + a, X_2 \le z + a, ..., X_n \le z + a) \phi(z) \,dz \\[6pt] &= \int \limits_{-\infty}^{\infty}\mathbb{P}(X_1 \le z + a) \cdot \mathbb{P}(X_2 \le z + a) \cdots \mathbb{P}(X_n \le z + a) \phi(z) \,dz \\[6pt] &= \int \limits_{-\infty}^{\infty}\Phi(z + a)^n \phi(z) \,dz. \\[6pt] \end{align}
Intuitively, $$z$$ (the variable of integration) is a scalar as opposed to a vector because it is the same variable that all $$X_1,...,X_n$$ are being compared with, and it allows us to compute the total probability utilizing the conditional probability given $$Z=z$$, which is much easier probability to compute. This alternative, easier formulation is allowed as per the law of total probability.
Observe that the integration variable would instead be the vector $$(z_1,...,z_n$$) if we were instead computing $$\mathbb{P}(X_1 \le Z_1 + a, X_2 \le Z_2 + a, ..., X_n \le Z_n + a.)$$