Biconditional Elimination/Sequent Form
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Theorem
\(\text {(1)}: \quad\) | \(\ds p \iff q\) | \(\vdash\) | \(\ds p \implies q\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds p \iff q\) | \(\vdash\) | \(\ds q \implies p\) |
Proof 1
Form 1
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \iff q$ | Premise | (None) | ||
2 | 1 | $p \implies q$ | Biconditional Elimination: $\iff \EE_1$ | 1 |
$\blacksquare$
Form 2
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \iff q$ | Premise | (None) | ||
2 | 1 | $q \implies p$ | Biconditional Elimination: $\iff \EE_2$ | 1 |
$\blacksquare$
Proof by Truth Table
We apply the Method of Truth Tables.
$\begin{array}{|ccc||ccc|ccc|} \hline p & \iff & q & p & \implies & q & q & \implies & p \\ \hline \F & \T & \F & \F & \T & \F & \F & \T & \F \\ \F & \F & \T & \F & \T & \T & \T & \F & \F \\ \T & \F & \F & \T & \F & \F & \F & \T & \T \\ \T & \T & \T & \T & \F & \T & \T & \T & \T \\ \hline \end{array}$
As can be seen, when $p \iff q$ is true so are both $p \implies q$ and $q \implies p$.
$\blacksquare$
Sources
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.5$: An aside: proof by contradiction: Exercises $1.6: \ 7$