# Proof by Cases/Sequent Form/Proof 1

## Theorem

Proof by Cases can be symbolised by the sequent:

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

## Proof

By the tableau method of natural deduction:

$p \lor q, \left({p \vdash r}\right), \left({q \vdash r}\right) \vdash r$
Line Pool Formula Rule Depends upon Notes
1 1 $p \lor q$ Premise (None)
2 2 $p$ Assumption (None)
3 2 $r$ By hypothesis 2 as $p \vdash r$
4 4 $q$ Assumption (None)
5 4 $r$ By hypothesis 4 as $q \vdash r$
6 1 $r$ Proof by Cases: $\text{PBC}$ 1 , 2 – 3 , 4 – 5 Assumptions 2  and 4 have been discharged

$\blacksquare$