# Biconditional is Commutative/Formulation 2

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## Theorems

The biconditional operator is commutative:

- $\vdash \left({p \iff q}\right) \iff \left({q \iff p}\right)$

## Proof

By the tableau method of natural deduction:

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

1 | 1 | $\left({p \iff q}\right)$ | Assumption | (None) | ||

2 | 1 | $\left({q \iff p}\right)$ | Sequent Introduction | 1 | Biconditional is Commutative: Formulation 1 | |

3 | 1 | $\left({p \iff q}\right) \implies \left({q \iff p}\right)$ | Rule of Implication: $\implies \mathcal I$ | 1 – 2 | Assumption 1 has been discharged | |

4 | 4 | $\left({p \iff q}\right)$ | Assumption | (None) | ||

5 | 4 | $\left({q \iff p}\right)$ | Sequent Introduction | 4 | Biconditional is Commutative: Formulation 1 | |

6 | 4 | $\left({p \iff q}\right) \implies \left({q \iff p}\right)$ | Rule of Implication: $\implies \mathcal I$ | 4 – 5 | Assumption 4 has been discharged | |

7 | $\left({p \iff q}\right) \iff \left({q \iff p}\right)$ | Biconditional Introduction: $\iff \mathcal I$ | 3, 6 |

$\blacksquare$

## Sources

- 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Theorem $\text{T92}$