Equivalences are Interderivable/Reverse Implication
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Theorem
If two propositional formulas are interderivable, they are equivalent:
- $\left ({p \iff q}\right) \vdash \left ({p \dashv \vdash q}\right)$
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \iff q$ | Premise | (None) | ||
3 | 1 | $p \implies q$ | Biconditional Elimination: $\iff \EE_1$ | 1 | ||
3 | 3 | $p$ | Assumption | (None) | ||
4 | 1, 3 | $q$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 2, 3 |
$\Box$
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \iff q$ | Premise | (None) | ||
3 | 1 | $p \implies q$ | Biconditional Elimination: $\iff \EE_2$ | 1 | ||
3 | 3 | $q$ | Assumption | (None) | ||
4 | 1, 3 | $p$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 2, 3 |
$\blacksquare$