Implication is Left Distributive over Disjunction/Formulation 1/Forward Implication

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Theorem

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


Proof

By the tableau method of natural deduction:

$p \implies \left({q \lor r}\right) \vdash \left({p \implies q}\right) \lor \left({p \implies r}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies \left({q \lor r}\right)$ Assumption (None)
2 2 $p$ Assumption (None)
3 1, 2 $q \lor r$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 2
4 2 $p$ Law of Identity 2
5 5 $q$ Assumption (None)
6 5 $p \implies q$ Rule of Implication: $\implies \mathcal I$ 4 – 5 Assumption 4 has been discharged
7 5 $\left({p \implies q}\right) \lor \left({p \implies r}\right)$ Rule of Addition: $\lor \mathcal I_1$ 6
8 8 $r$ Assumption (None)
9 8 $p \implies r$ Sequent Introduction 8 True Statement is implied by Every Statement
10 8 $\left({p \implies q}\right) \lor \left({p \implies r}\right)$ Rule of Addition: $\lor \mathcal I_2$ 9
11 1 $\left({p \implies q}\right) \lor \left({p \implies r}\right)$ Proof by Cases: $\text{PBC}$ 3, 2 – 7, 8 – 10 Assumptions 2 and 8 have been discharged

$\blacksquare$