Theorem 3anidm 1101

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

Assertion

Ref	Expression
3anidm	⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) ↔ 𝜑)

Proof of Theorem 3anidm

Step	Hyp	Ref	Expression
1		df-3an 1086	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) ↔ ((𝜑 ∧ 𝜑) ∧ 𝜑))
2		anabs1 661	. 2 ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜑) ↔ (𝜑 ∧ 𝜑))
3		anidm 568	. 2 ⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)
4	1, 2, 3	3bitri 300	1 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) ↔ 𝜑)

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086

This theorem is referenced by:  relcnvtr  6097