Theorem in32 4216

Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
in32	⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ 𝐵)

Proof of Theorem in32

Step	Hyp	Ref	Expression
1		inass 4214	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶))
2		in12 4215	. 2 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶))
3		incom 4196	. 2 ⊢ (𝐵 ∩ (𝐴 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ 𝐵)
4	1, 2, 3	3eqtri 2763	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ 𝐵)

Syntax hints:   = wceq 1541   ∩ cin 3942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3474  df-in 3950

This theorem is referenced by:  in13  4217  inrot  4219  wefrc  5662  imainrect  6168  sspred  6297  fpwwe2  10619  incexclem  15763  setsfun  17085  setsfun0  17086  ressress  17174  kgeni  22967  kgencn3  22988  fclsrest  23454  voliunlem1  24993  bj-disj2r  35699  refrelsredund4  37293