化简 $$$\overline{\left(\overline{A} + B\right) \cdot \left(\overline{B} + C\right)}$$$

对 $$$x = \overline{A} + B$$$ 和 $$$y = \overline{B} + C$$$ 应用德摩根定律 $$$\overline{x \cdot y} = \overline{x} + \overline{y}$$$：

$${\color{red}\left(\overline{\left(\overline{A} + B\right) \cdot \left(\overline{B} + C\right)}\right)} = {\color{red}\left(\overline{\overline{A} + B} + \overline{\overline{B} + C}\right)}$$

对 $$$x = \overline{A}$$$ 和 $$$y = B$$$ 应用德摩根定律 $$$\overline{x + y} = \overline{x} \cdot \overline{y}$$$：

$${\color{red}\left(\overline{\overline{A} + B}\right)} + \overline{\overline{B} + C} = {\color{red}\left(\overline{\overline{A}} \cdot \overline{B}\right)} + \overline{\overline{B} + C}$$

对 $$$x = A$$$ 应用双重否定（对合）律 $$$\overline{\overline{x}} = x$$$：

$$\left({\color{red}\left(\overline{\overline{A}}\right)} \cdot \overline{B}\right) + \overline{\overline{B} + C} = \left({\color{red}\left(A\right)} \cdot \overline{B}\right) + \overline{\overline{B} + C}$$

对 $$$x = \overline{B}$$$ 和 $$$y = C$$$ 应用德摩根定律 $$$\overline{x + y} = \overline{x} \cdot \overline{y}$$$：

$$\left(A \cdot \overline{B}\right) + {\color{red}\left(\overline{\overline{B} + C}\right)} = \left(A \cdot \overline{B}\right) + {\color{red}\left(\overline{\overline{B}} \cdot \overline{C}\right)}$$

对 $$$x = B$$$ 应用双重否定（对合）律 $$$\overline{\overline{x}} = x$$$：

$$\left(A \cdot \overline{B}\right) + \left({\color{red}\left(\overline{\overline{B}}\right)} \cdot \overline{C}\right) = \left(A \cdot \overline{B}\right) + \left({\color{red}\left(B\right)} \cdot \overline{C}\right)$$