Theorem 3anan12

Description: Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (Proof shortened by Andrew Salmon, 14-Jun-2011) (Revised to shorten 3ancoma by Wolf Lammen, 5-Jun-2022.)

	Ref	Expression
Assertion	3anan12	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\psi \wedge \left(\phi \wedge \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		3anass	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\phi \wedge \left(\psi \wedge \chi \right)\right)$$
2		an12	$$⊢\left(\phi \wedge \left(\psi \wedge \chi \right)\right)\leftrightarrow \left(\psi \wedge \left(\phi \wedge \chi \right)\right)$$
3	1 2	bitri	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\psi \wedge \left(\phi \wedge \chi \right)\right)$$