Rule of Idempotence/Conjunction/Formulation 2

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Theorem

The conjunction operator is idempotent:

$\vdash p \iff \paren {p \land p}$


Proof

By the tableau method of natural deduction:

$\vdash p \iff \paren {p \land p} $
Line Pool Formula Rule Depends upon Notes
1 1 $p$ Assumption (None)
2 1 $p \land p$ Rule of Conjunction: $\land \mathcal I$ 1, 1
3 $p \implies \paren {p \land p}$ Rule of Implication: $\implies \mathcal I$ 1 – 2 Assumption 1 has been discharged
4 4 $p \land p$ Assumption (None)
5 4 $p$ Rule of Simplification: $\land \mathcal E_1$ 4
6 $\paren {p \land p} \implies p$ Rule of Implication: $\implies \mathcal I$ 4 – 5 Assumption 4 has been discharged
7 $p \iff \paren {p \land p}$ Biconditional Introduction: $\iff \mathcal I$ 3, 6

$\blacksquare$


Sources