Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Reverse Implication

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Definition

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


Proof

By the tableau method of natural deduction:

$\left({p \land q}\right) \lor \left({p \land r}\right) \vdash p \land \left({q \lor r}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $\left({p \land q}\right) \lor \left({p \land r}\right)$ Premise (None)
2 2 $p \land q$ Assumption (None)
3 2 $p$ Rule of Simplification: $\land \mathcal E_1$ 2
4 4 $p \land r$ Assumption (None)
5 4 $p$ Rule of Simplification: $\land \mathcal E_1$ 2
6 1 $p$ Proof by Cases: $\text{PBC}$ 1, 2 – 3, 4 – 5 Assumptions 2 and 4 have been discharged
7 7 $p \land q$ Assumption (None)
8 7 $q$ Rule of Simplification: $\land \mathcal E_2$ 7
9 7 $q \lor r$ Rule of Addition: $\lor \mathcal I_1$ 7
10 10 $p \land r$ Assumption (None)
11 10 $r$ Rule of Simplification: $\land \mathcal E_2$ 10
12 10 $q \lor r$ Rule of Addition: $\lor \mathcal I_2$ 11
13 1 $q \lor r$ Proof by Cases: $\text{PBC}$ 1, 7 – 9, 10 – 12 Assumptions 7 and 10 have been discharged
14 1 $p \land \left({q \lor r}\right)$ Rule of Conjunction: $\land \mathcal I$ 6, 13


$\blacksquare$


Sources