Rule of Commutation/Conjunction/Formulation 2

Theorem

$\vdash \paren {p \land q} \iff \paren {q \land p}$

$\vdash \paren {p \land q} \iff \paren {q \land p} $
Line	Pool	Formula	Rule	Depends upon	Notes
1	1	$p \land q$	Assumption	(None)
2	1	$q \land p$	Sequent Introduction	1	Conjunction is Commutative
3		$\paren {p \land q} \implies \paren {q \land p}$	Rule of Implication: $\implies \II$	1 – 2	Assumption 1 has been discharged
4	4	$q \land p$	Assumption	(None)
5	4	$p \land q$	Sequent Introduction	4	Conjunction is Commutative
6		$\paren {q \land p} \implies \paren {p \land q}$	Rule of Implication: $\implies \II$	4 – 5	Assumption 4 has been discharged
7		$\paren {p \land q} \iff \paren {q \land p}$	Biconditional Introduction: $\iff \II$	3, 6

$\blacksquare$

1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.13$: Symbolism of sentential calculus
1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 3.6$: Reference Formulae: $RF \, 8$
1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 4.7$: The Derivation of Formulae: $D \, 21 - 22$
1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$: Theorem $\text{T24}$
1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.2$: The Rule of Replacement: $11.$
1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 6$: Using logical equivalences: $10$
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(1) \ \text{(iii)}$
1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.14$: Exercise $12 \ (4)$