Theorem in13 4191

Description: A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)

Assertion

Ref	Expression
in13	⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴))

Proof of Theorem in13

Step	Hyp	Ref	Expression
1		in32 4190	. 2 ⊢ ((𝐵 ∩ 𝐶) ∩ 𝐴) = ((𝐵 ∩ 𝐴) ∩ 𝐶)
2		incom 4170	. 2 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐵 ∩ 𝐶) ∩ 𝐴)
3		incom 4170	. 2 ⊢ (𝐶 ∩ (𝐵 ∩ 𝐴)) = ((𝐵 ∩ 𝐴) ∩ 𝐶)
4	1, 2, 3	3eqtr4i 2802	1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴))

Syntax hints:   = wceq 1567   ∩ cin 3912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-in 3920

This theorem is referenced by:  inin  32802