Rule of Association/Conjunction/Formulation 1/Proof 1

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Theorem

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


Proof

By the tableau method of natural deduction:

$p \land \left({q \land r}\right) \vdash \left({p \land q}\right) \land r$
Line Pool Formula Rule Depends upon Notes
1 1 $p \land \left({q \land r}\right)$ Premise (None)
2 1 $p$ Rule of Simplification: $\land \EE_1$ 1
3 1 $q \land r$ Rule of Simplification: $\land \EE_2$ 1
4 1 $q$ Rule of Simplification: $\land \EE_1$ 3
5 1 $r$ Rule of Simplification: $\land \EE_2$ 3
6 1 $p \land q$ Rule of Conjunction: $\land \II$ 2, 4
7 1 $\left({p \land q}\right) \land r$ Rule of Conjunction: $\land \II$ 6, 5


By the tableau method of natural deduction:

$\left({p \land q}\right) \land r \vdash p \land \left({q \land r}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $\left({p \land q}\right) \land r$ Premise (None)
2 1 $p \land q$ Rule of Simplification: $\land \EE_1$ 1
3 1 $r$ Rule of Simplification: $\land \EE_2$ 1
4 1 $p$ Rule of Simplification: $\land \EE_1$ 2
5 1 $q$ Rule of Simplification: $\land \EE_2$ 2
6 1 $q \land r$ Rule of Conjunction: $\land \II$ 5, 3
7 1 $p \land \left({q \land r}\right)$ Rule of Conjunction: $\land \II$ 4, 6

$\blacksquare$