公式的基本等价关系（数理逻辑）

定理 1　公式的基本等价关系

　　 令 $A,B,C$ 都是任意的合式公式，则有以下的关系成立：

1. 幂等律：
   $$\begin{array}{}\text{(1)}& \begin{array}{rl}A\wedge A& \iff A,\\ A\vee A& \iff A.\end{array}\end{array}$$
2. 交换律：
   $$\begin{array}{}\text{(2)}& \begin{array}{rl}A\wedge B& \iff B\wedge A,\\ A\vee B& \iff B\vee A.\end{array}\end{array}$$
3. 结合律：
   $$\begin{array}{}\text{(3)}& \begin{array}{rl}A\wedge (B\wedge C)& \iff (A\wedge B)\wedge C,\\ A\vee (B\vee C)& \iff (A\vee B)\vee C.\end{array}\end{array}$$
4. 同一律：
   $$\begin{array}{}\text{(4)}& \begin{array}{rl}A\wedge 1& \iff A,\\ A\vee 0& \iff A.\end{array}\end{array}$$
5. “分配律”：
   $$\begin{array}{}\text{(5)}& \begin{array}{rl}A\vee (B\wedge C)& \iff (A\vee B)\wedge (A\vee C),\\ A\wedge (B\vee C)& \iff (A\wedge B)\vee (A\wedge C).\end{array}\end{array}$$
6. 逆否（假言易位）：
   $$\begin{array}{}\text{(6)}& (A\to B)\iff ((\mathrm{\neg }B)\to (\mathrm{\neg }A)).\end{array}$$
7. 吸收率：
   $$\begin{array}{}\text{(7)}& \begin{array}{rl}A\vee (A\wedge B)& \iff A,\\ A\wedge (A\vee B)& \iff A.\end{array}\end{array}$$
8. 矛盾率与排中律：
   $$\begin{array}{}\text{(8)}& \begin{array}{rl}A\wedge (\mathrm{\neg }A)& \iff 0,\\ A\vee (\mathrm{\neg }A)& \iff 1.\end{array}\end{array}$$
9. 双重否定：
   $$\begin{array}{}\text{(9)}& \mathrm{\neg }(\mathrm{\neg }A)\iff A.\end{array}$$
10. 德摩根定率（De Morgan's laws）：
    $$\begin{array}{}\text{(10)}& \begin{array}{rl}\mathrm{\neg }(A\vee B)& \iff (\mathrm{\neg }A)\wedge (\mathrm{\neg }B),\\ \mathrm{\neg }(A\wedge B)& \iff (\mathrm{\neg }A)\vee (\mathrm{\neg }B).\end{array}\end{array}$$
11. 蕴含：
    $$\begin{array}{}\text{(11)}& A\to B\iff (\mathrm{\neg }A)\vee B.\end{array}$$
12. 等价：
    $$\begin{array}{}\text{(12)}& (A\leftrightarrow B)\iff ((A\to B)\wedge (B\to A)).\end{array}$$