Rule of Commutation/Conjunction/Formulation 1/Proof 1

Theorem

$p \land q \dashv \vdash q \land p$

Proof

By the tableau method of natural deduction:

$p \land q \vdash q \land p$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p \land q$	Premise	(None)
2		1	$p$	Rule of Simplification: $\land \EE_1$	1
3		1	$q$	Rule of Simplification: $\land \EE_2$	1
4		1	$q \land p$	Rule of Conjunction: $\land \II$	3, 2

$\Box$

By the tableau method of natural deduction:

$q \land p \vdash p \land q$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$q \land p$	Premise	(None)
2		1	$q$	Rule of Simplification: $\land \EE_1$	1
2		1	$p$	Rule of Simplification: $\land \EE_2$	1
4		1	$p \land q$	Rule of Conjunction: $\land \II$	3, 2

$\blacksquare$

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $3$ Conjunction and Disjunction: Theorem $17$