# Proof by Cases/Explanation

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## Proof Rule

Proof by Cases can be expressed in natural language as follows:

We are given that either $\phi$ is true, or $\psi$ is true, or both.

Suppose we make the assumption that $\phi$ is true, and from that deduce that $\chi$ has to be true.

Then suppose we make the assumption that $\psi$ is true, and from that deduce that $\chi$ has to be true.

Therefore, it has to follow that the truth of $\chi$ follows from the fact of the truth of either $\phi$ or $\psi$.