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

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Theorem

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


Proof

By the tableau method of natural deduction:

$p \vdash p \land \left({p \lor q}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $p$ Premise (None)
2 1 $p \lor q$ Rule of Addition: $\lor \mathcal I_1$ 1
3 1 $p \land \left({p \lor q}\right)$ Rule of Conjunction: $\land \mathcal I$ 1, 2

$\blacksquare$


Sources