Theorem anandi3 1101

Description: Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)

Assertion

Ref	Expression
anandi3	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))

Proof of Theorem anandi3

Step	Hyp	Ref	Expression
1		3anass 1094	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
2		anandi 673	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
3	1, 2	bitri 274	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1086

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 206  df-an 397  df-3an 1088

This theorem is referenced by:  ndmovdistr  7461  cusgr3cyclex  33098