structural:reformatting, quality:correction
Łukasiewicz's P

Jan Łukasiewicz showed that, in Frege's system, "the third axiom is superfluous since it can be derived from the preceding two axioms, and that the last three axioms can be replaced by the single sentence C C N p N q C p q {\displaystyle CCNpNqCpq} ". Which, taken out of Łukasiewicz's Polish notation into modern notation, means (¬ p → ¬ q) → (p → q) {\displaystyle (\neg p\rightarrow \neg q)\rightarrow (p\rightarrow q)}. Hence, Łukasiewicz is credited with this system of three axioms:

Just like Frege's system, this system uses a substitution rule and uses modus ponens as an inference rule. The exact same system was given (with an explicit substitution rule) by Alonzo Church, who referred to it as the system P and helped popularize it.