Axioms

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

Rules

R1.

$\vdash A$

$\vdash A \rightarrow B$

$\vdash B$ Modus Ponens

R2.

$\vdash A$

$\vdash B$

$\vdash A \land B$ Adjunction