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

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Theorem

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


Proof

By the tableau method of natural deduction:

$p \vdash p \land \paren {p \lor q} $
Line Pool Formula Rule Depends upon Notes
1 1 $p$ Premise (None)
2 1 $p \lor q$ Rule of Addition: $\lor \II_1$ 1
3 1 $p \land \paren {p \lor q}$ Rule of Conjunction: $\land \II$ 1, 2

$\blacksquare$


Sources