Biconditional is Commutative/Formulation 1/Proof 2

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Proof

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

$\blacksquare$


Sources