# Biconditional with Tautology/Proof 1

## Theorem

$p \iff \top \dashv \vdash p$

## Proof

By the tableau method of natural deduction:

$p \iff \top \vdash p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \iff \top$ Premise (None)
2 $\top$ Rule of Top-Introduction: $\top \mathcal I$ (None)
3 1 $\top \implies p$ Biconditional Elimination: $\iff \mathcal E_2$ 1
4 1 $p$ Modus Ponendo Ponens: $\implies \mathcal E$ 2, 3

$\Box$

By the tableau method of natural deduction:

$p \vdash p \iff \top$
Line Pool Formula Rule Depends upon Notes
1 1 $\top$ Premise (None)
2 2 $p$ Assumption (None)
3 $\top$ Rule of Top-Introduction: $\top \mathcal I$ (None)
4 $p \implies \top$ Rule of Implication: $\implies \mathcal I$ 2 – 3 Assumption 2 has been discharged
5 2 $\top \implies p$ Rule of Implication: $\implies \mathcal I$ 1 – 2 Assumption 1 has been discharged
6 2 $p \iff \top$ Biconditional Introduction: $\iff \mathcal I$ 4, 5

$\blacksquare$