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 # ECE 208: Discrete Math 2
 
+## Hilbert system rules
 
+**Axioms:**
+
+- $\vdash (A\implies (B\implies A))$
+- $\vdash (A\implies (B\implies C))\implies ((A\implies B)\implies (A\implies C))$
+- $\vdash (\neg B\implies\neg A)\implies (A\implies B))$
+
+**Inference (MP):**
+
+- $\frac{\vdash A, \vdash A\implies B}{\vdash B}$
+
+**Derived rules:**
+
+- Deduction: $\frac{U\cup \{A\}\vdash B}{U\vdash A\implies B}$
+- Contrapositive: $\frac{U\vdash \neg B\implies\neg A}{U\vdash A\implies B}$ (and vice versa)
+- Transitivity: $\frac{u\vdash A\implies B, U\vdash B\implies C}{U\vdash A\implies C}$
+- Exchange of antecedent: $\frac{U\vdash A\implies (B\implies C)}{U\vdash B\implies (A\implies C)}$
+- Double negation: $\frac{U\vdash \neg\neg A}{U\vdash A}$ (and vice versa)
+- Reductio ad absurdum: $\frac{U\vdash\neg A\implies false}{U\vdash A}$