Biconditional Equivalent to Biconditional of Negations/Formulation 2
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Theorem
- $\vdash \left({p \iff q}\right) \iff \left({\neg p \iff \neg q}\right)$
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \iff q$ | Assumption | (None) | ||
2 | 1 | $\neg p \iff \neg q$ | Sequent Introduction | 1 | Biconditional Equivalent to Biconditional of Negations: Formulation 1 | |
3 | $\left({p \iff q}\right) \implies \left({\neg p \iff \neg q}\right)$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged | ||
4 | 4 | $\neg p \iff \neg q$ | Assumption | (None) | ||
5 | 4 | $p \iff q$ | Sequent Introduction | 4 | Biconditional Equivalent to Biconditional of Negations: Formulation 1 | |
6 | $\left({\neg p \iff \neg q}\right) \implies \left({p \iff q}\right)$ | Rule of Implication: $\implies \II$ | 4 – 5 | Assumption 4 has been discharged | ||
7 | $\left({p \iff q}\right) \iff \left({\neg p \iff \neg q}\right)$ | Biconditional Introduction: $\iff \II$ | 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 $\text{T96}$