# Law of Identity/Formulation 2/Proof 1

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## Theorem

- $\vdash p \implies p$

## Proof

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $p$ | Premise | (None) | ||

2 | $p \implies p$ | Rule of Implication: $\implies \mathcal I$ | 1 – 1 | Assumption 1 has been discharged |

$\blacksquare$

This is the second shortest tableau proof possible.

## Sources

- 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 5$: Theorem $\text{T1}$ - 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 2.2$: Theorems and Derived Rules: Theorem $38$ - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction