Absorption Laws (Logic)/Disjunction Absorbs Conjunction/Forward Implication

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Theorem

$p \lor \paren {p \land q} \vdash p$


Proof

By the tableau method of natural deduction:

$p \lor \paren {p \land q} \vdash p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \lor \paren {p \land q}$ Premise (None)
2 2 $p$ Assumption (None)
3 3 $p \land q$ Assumption (None)
4 3 $p$ Rule of Simplification: $\land \mathcal E_1$ 3
5 1 $p$ Proof by Cases: $\text{PBC}$ 1, 2 – 2, 3 – 4 Assumptions 2 and 3 have been discharged

$\blacksquare$


Sources