Theorem 3anidm

Description: Idempotent law for conjunction. (Contributed by Peter Mazsa, 17-Oct-2023)

	Ref	Expression
Assertion	3anidm	$$⊢\left(\phi \wedge \phi \wedge \phi \right)\leftrightarrow \phi$$

Proof

Step	Hyp	Ref	Expression
1		df-3an	$$⊢\left(\phi \wedge \phi \wedge \phi \right)\leftrightarrow \left(\left(\phi \wedge \phi \right)\wedge \phi \right)$$
2		anabs1	$$⊢\left(\left(\phi \wedge \phi \right)\wedge \phi \right)\leftrightarrow \left(\phi \wedge \phi \right)$$
3		anidm	$$⊢\left(\phi \wedge \phi \right)\leftrightarrow \phi$$
4	1 2 3	3bitri	$$⊢\left(\phi \wedge \phi \wedge \phi \right)\leftrightarrow \phi$$