I came across the following in Entailment vol. 1 trying to get unstuck on some stuff. It is, quite possibly, the snappiest way of putting a theorem of logic I’ve heard. A possible contender is Halmos’s way of putting the crowning glory of modern logic. Here is the theorem:
Manifest repugnancies entail every truth function to which they are analytically relevant.
Anderson and Belnap, being logicians, define manifest repugnancies and being analytically relevant. The former is a conjunction of formulas such that for all atomic formulas p occurring in it, ~p also occurs in it. The latter is defined as A is analytically relevant to B if all propositional variables of B occur in A. This is within the context of the logic E, I think.