Biconditional is Commutative/Formulation 1

Theorems

$p \iff q \dashv \vdash q \iff p$

Proof 1

By the tableau method of natural deduction:

$p \iff q \vdash q \iff p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \iff q$ Premise (None)
2 1 $p \implies q$ Biconditional Elimination: $\iff \mathcal E_1$ 1
3 1 $q \implies p$ Biconditional Elimination: $\iff \mathcal E_2$ 1
4 1 $q \iff p$ Biconditional Introduction: $\iff \mathcal I$ 3, 2

$\Box$

By the tableau method of natural deduction:

$q \iff p \vdash p \iff q$
Line Pool Formula Rule Depends upon Notes
1 1 $q \iff p$ Premise (None)
2 1 $q \implies p$ Biconditional Elimination: $\iff \mathcal E_1$ 1
3 1 $p \implies q$ Biconditional Elimination: $\iff \mathcal E_2$ 1
4 1 $p \iff q$ Biconditional Introduction: $\iff \mathcal I$ 3, 2

$\blacksquare$

Proof 2

By the tableau method of natural deduction:

$p \iff q \vdash q \iff p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \iff q$ Premise (None)
2 1 $\left({p \implies q}\right) \land \left({q \implies p}\right)$ Sequent Introduction 1 Rule of Material Equivalence
3 1 $\left({q \implies p}\right) \land \left({p \implies q}\right)$ Sequent Introduction 2 Conjunction is Commutative
4 1 $q \iff p$ Sequent Introduction 3 Rule of Material Equivalence

$\Box$

By the tableau method of natural deduction:

$q \iff p \vdash p \iff q$
Line Pool Formula Rule Depends upon Notes
1 1 $q \iff p$ Premise (None)
2 1 $\left({q \implies p}\right) \land \left({p \implies q}\right)$ Sequent Introduction 1 Rule of Material Equivalence
3 1 $\left({p \implies q}\right) \land \left({q \implies p}\right)$ Sequent Introduction 2 Conjunction is Commutative
4 1 $p \iff q$ Sequent Introduction 3 Rule of Material Equivalence

$\blacksquare$

Proof 3

$p \iff q \dashv \vdash q \iff p$

We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.

$\begin{array}{|ccc||ccc|} \hline p & \iff & q & q & \iff & p \\ \hline F & T & F & F & T & F \\ F & F & T & T & F & F \\ T & F & F & F & F & T \\ T & T & T & T & T & T \\ \hline \end{array}$

$\blacksquare$