Proof by Cases

Context
This is one of the axioms of natural deduction.

The rule
If we can conclude $$p \lor q$$, and:


 * 1) By making the assumption $$p$$, we can conclude $$r$$;
 * 2) By making the assumption $$q$$, we can conclude $$r$$;

then we may infer $$r$$.

$$p \lor q, \left({p \vdash r}\right), \left({q \vdash r}\right) \vdash r$$

The conclusion does not depend upon either assumption $$p$$ or $$q$$.


 * Abbreviation: $$\lor \mathcal{E}$$
 * Deduced from: The pooled assumptions of:
 * 1) The instance of $$p \lor q$$;
 * 2) The instance of $$r$$ which was derived from the individual assumption $$p$$;
 * 3) The instance of $$r$$ which was derived from the individual assumption $$q$$.


 * Discharged Assumptions: The assumptions $$p$$ and $$q$$.
 * Depends on: The following three things:
 * 1) The line containing the instance of $$p \lor q$$;
 * 2) The series of lines from where the assumption $$p$$ was made to where $$r$$ was deduced;
 * 3) The series of lines from where the assumption $$q$$ was made to where $$r$$ was deduced.

Explanation
We know $$p \lor q$$, that is, either $$p$$ is true or $$q$$ is true, or both.

Suppose we assume that $$p$$ is true, and from that assumption we have managed to deduce that $$r$$ has to be true.

Then suppose we assume that $$q$$ is true, and from that assumption we have also managed to deduce that $$r$$ has to be true.

Therefore, it has to follow that the truth of $$r$$ follows from the fact of the truth of $$p \lor q$$.

Thus we can eliminate a disjunction from a sequent.