Is the equivalence true? $$\left( \forall_{i}: x \leqslant i \rightarrow y \leqslant i \right) \iff y \leqslant x$$
I left the universe of discourse blank since I guess this makes no difference, but it's fine to assume all variables on $\mathbb Z$.
The $\Leftarrow$ part is quite easy since it follows from transitivity.
I guess the $\Rightarrow$ part can be proven by cases, but such a apparently simple proposition requiring a proof by cases seems overkill.
If you can give me a reference for your theory would be nice.