Class Comprehension in Two and Three Set Variables, Stated with Ordered Pairs

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation. This is a theorem scheme of the metatheory: it asserts one helper for each formula as below.

Let $u$, $v$ and $w$ be distinct set variables, and let $Y_{1},\dots,Y_{m}$, with $m\ge0$ a metatheoretic number, be distinct variables of either kind, different from $u$, $v$ and $w$ (the parameters), with a set assigned to each parameter that is a set variable as in the reading of formulas. For sets $u$ and $v$, $(u,v)$ denotes their ordered pair.

(i) If $\varphi(u,v,Y_{1},\dots,Y_{m})$ is a predicative formula whose free variables are among $u,v,Y_{1},\dots,Y_{m}$, then for all classes $Y_{1},\dots,Y_{m}$ there is a class $Z$ such that, for all sets $u$ and $v$,

$$(u,v)\in Z\quad\text{if and only if}\quad \varphi(u,v,Y_{1},\dots,Y_{m}).$$

(ii) If $\varphi(u,v,w,Y_{1},\dots,Y_{m})$ is a predicative formula whose free variables are among $u,v,w,Y_{1},\dots,Y_{m}$, then for all classes $Y_{1},\dots,Y_{m}$ there is a class $Z$ such that, for all sets $u$, $v$ and $w$,

$$((u,v),w)\in Z\quad\text{if and only if}\quad \varphi(u,v,w,Y_{1},\dots,Y_{m}).$$