# Rule of Material Implication/Formulation 1/Reverse Implication/Proof 1

## Theorem

$\neg p \lor q \vdash p \implies q$

## Proof

By the tableau method of natural deduction:

$\neg p \lor q \vdash p \implies q$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg p \lor q$ Premise (None)
2 2 $\neg p$ Assumption (None) Pick the first of the disjuncts ...
3 3 $p$ Assumption (None) Assume its negation ...
4 2, 3 $\bot$ Principle of Non-Contradiction: $\neg \EE$ 3, 2 ... and demonstrate a contradiction
5 2, 3 $q$ Rule of Explosion: $\bot \EE$ 4 ... from a falsehood, any statement can be derived - pick $q$
6 2 $p \implies q$ Rule of Implication: $\implies \II$ 3 – 5 Assumption 3 has been discharged
7 7 $q$ Assumption (None) Pick the second of the disjuncts ...
8 8 $p$ Assumption (None) ... again assume $p$ ...
9 7 $q$ Law of Identity 7 The truth of $q$ still holds
10 7 $p \implies q$ Rule of Implication: $\implies \II$ 8 – 9 Assumption 8 has been discharged
11 1 $p \implies q$ Proof by Cases: $\text{PBC}$ 1, 2 – 6, 7 – 10 Assumptions 2 and 7 have been discharged

$\blacksquare$