Theorem anandi3r 1100

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

Assertion

Ref	Expression
anandi3r	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)))

Proof of Theorem anandi3r

Step	Hyp	Ref	Expression
1		3anan32 1094	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓))
2		anandir 675	. 2 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)))
3	1, 2	bitri 274	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∧ 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 206  df-an 395  df-3an 1086

This theorem is referenced by:  refsymrel2  38071  refsymrel3  38072  dfeqvrel2  38094  dfeqvrel3  38095  i0oii  48016  alsi-no-surprise  48307