Exotic schemes of implications, examples

Question

We are all familiar with schemes of implication like:
$A\rightarrow B $
$A\iff B $

Or even more complex structures like a collection of three statements any two of which imply the third one.

Are there very unusual schemes of implication you are aware have ever been used? Eg. a collection of five statements any three of which implies one and only one of the other two.

To be clearer, I'm not asking whether those schemes are logically possible, because they certainly are,for instance the previous example would be expressed as:
Let $S=\{A_i\}_{i=0,1,...,4} $ be a collection of five statements. $\forall (A_j,A_k,A_l,A_m,A_n) \in S^5 s.t.j,k,l,m,n$ are all different $(A_j\wedge A_k \wedge A_l)\rightarrow (A_m\veebar A_n) $.

I'm asking if there are examples of theorems or proves that have such structures. For example "at least one of $e\pi $ and $e+\pi $ is transcendental" is a quite unusual theorem.

It may seem a bizarre question but it is about a deeper question in mathematics. As humans we strive for symmetry and the theorems we produce are thus actually very symmetrical/simple statements (compared to the complexity of the field in which they occur) that are also dense of information, theorems are like landmarks that help us finding the right path among an infinite number of paths. Yet, by preferring these simple statements we may be overlooking important theorems just because they are too convoluted.

Now, I do believe that any such more complex theorem could be split into simpler theorems but, of course, I cannot be sure of that, so I wanted to find theorems or proofs with stranger implication schemes, to have more material to work on.

Answer 1

I guess that every classification theorem falls along the lines of being essentially and or result (stronger though, since the or is exclusive).

For example, the fundamental theorem of lineal programming can be stated as follows:

A linear programming problem has an optimal basic feasible solution xor has an unbounded optimum xor is unfeasible

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There is also this proposition in algebraic equations where if $f\in \mathbb{R}[T]$ irreducible and solvable is of degree $p$ where $p\ge 3$ is prime, then $f$ has $1$ or $p$ real roots, which kinda fits the implication schema you've given.

These are not particularly exciting examples, but I cannot think of good examples where the implication schema is complex and/or subtle rather than outright complicated.