Axioms

1. $\vdash A \rightarrow A$ Identity
2. $\vdash (A \wedge B) \rightarrow A$ Simplification
3. $\vdash (A \wedge B) \rightarrow B$ Simplification
4. $\vdash A \rightarrow (A \lor B)$ Addition
5. $\vdash B \rightarrow (A \lor B)$ Addition
6. $\vdash (A \wedge (B \lor C)) \rightarrow ((A \wedge B) \lor C)$ Distribution
7. $\vdash ((A \rightarrow B) \land (A \rightarrow C)) \rightarrow (A \rightarrow (B \land C))$ Strong Lattice $\land$
8. $\vdash ((A \rightarrow C) \lor (B \rightarrow C)) \rightarrow ((A \lor B) \rightarrow C)$ Strong Lattice $\lor$
9. $\vdash \neg \neg A \rightarrow A$ Double Negation Elimination
10. $\vdash (A \rightarrow \neg B) \rightarrow (B \rightarrow \neg A)$ Contraposition
11. $\vdash (A \rightarrow B) \rightarrow ((B \rightarrow C) \rightarrow (A \rightarrow C))$ Suffixing
12. $\vdash ((A \rightarrow (A \rightarrow B)) \rightarrow (A \rightarrow B)$ Contraction
13. $\vdash ((A \rightarrow A) \rightarrow B) \rightarrow B$ Demodalizer

Rules

R1.

$\vdash A$

$\vdash A \rightarrow B$

$\vdash B$ Modus Ponens

R2.

$\vdash A$

$\vdash B$

$\vdash A \land B$ Adjunction