Equivalences are Interderivable/Forward Implication
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Theorem
If two propositional formulas are interderivable, then they are equivalent:
- $\left ({p \dashv \vdash q}\right) \vdash \left ({p \iff q}\right)$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \dashv \vdash q$ | Premise | (None) | ||
| 2 | 1 | $\left ({p \vdash q}\right) \land \left ({q \vdash p}\right)$ | Sequent Introduction | 1 | Definition of Interderivable | |
| 3 | 3 | $p$ | Assumption | (None) | ||
| 4 | 1, 3 | $p \vdash q$ | Rule of Simplification: $\land \EE_1$ | 2 | ||
| 5 | 1 | $p \implies q$ | Rule of Implication: $\implies \II$ | 3 – 4 | Assumption 3 has been discharged | |
| 6 | 6 | $q$ | Assumption | (None) | ||
| 7 | 1, 6 | $q \vdash p$ | Rule of Simplification: $\land \EE_2$ | 2 | ||
| 8 | 1 | $q \implies p$ | Rule of Implication: $\implies \II$ | 6 – 7 | Assumption 6 has been discharged | |
| 9 | 1 | $p \iff q$ | Biconditional Introduction: $\iff \II$ | 5, 8 |
$\blacksquare$