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## Main Question or Discussion Point

I am reading a pdf where, under a "classic ways of reasoning" section, the author introduced a method called

$$\text{Let }\mathrm A,\,\mathrm B\text{ and }\mathrm C\text{ be three propositions, then:}\\

\text{This implication is always true: } ((\mathrm A\Rightarrow\mathrm C)\wedge(\mathrm B\Rightarrow\mathrm C))\Rightarrow((\mathrm A \vee \mathrm B)\Rightarrow\mathrm C)$$

I am not sure I understand the point of this, here's how I am thinking about it:

If I can show that ##((\mathrm A\Rightarrow\mathrm C)\wedge(\mathrm B\Rightarrow\mathrm C))## is true, then I have proved that ##((\mathrm A \vee \mathrm B)\Rightarrow\mathrm C)## is true. I know nothing about the truth value of ##\mathrm C##, so I should prove that both ##\mathrm A## and ##\mathrm B## are true to force the truthfulness of the proposition on the left side of the main implication and thus that on the right.

I feel that I am missing something, though, or that I am not seeing the main point. If you could evaluate my reasoning, and/or add something more, I'd be grateful.

*la disjonction de cas*, which I think in English would be "case by case" reasoning. He enounced it as follows:$$\text{Let }\mathrm A,\,\mathrm B\text{ and }\mathrm C\text{ be three propositions, then:}\\

\text{This implication is always true: } ((\mathrm A\Rightarrow\mathrm C)\wedge(\mathrm B\Rightarrow\mathrm C))\Rightarrow((\mathrm A \vee \mathrm B)\Rightarrow\mathrm C)$$

I am not sure I understand the point of this, here's how I am thinking about it:

If I can show that ##((\mathrm A\Rightarrow\mathrm C)\wedge(\mathrm B\Rightarrow\mathrm C))## is true, then I have proved that ##((\mathrm A \vee \mathrm B)\Rightarrow\mathrm C)## is true. I know nothing about the truth value of ##\mathrm C##, so I should prove that both ##\mathrm A## and ##\mathrm B## are true to force the truthfulness of the proposition on the left side of the main implication and thus that on the right.

I feel that I am missing something, though, or that I am not seeing the main point. If you could evaluate my reasoning, and/or add something more, I'd be grateful.

**EDIT:**I should prove that the implications on both sides of the conjunction are true not ##\mathrm A## and ##\mathrm B##, since showing the latter propositions to be doesn't imply that the right side of the main implication is true as ##\mathrm C## might not follow.
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