Theorem in32 4197

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 4195	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶))
2		in12 4196	. 2 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶))
3		incom 4177	. 2 ⊢ (𝐵 ∩ (𝐴 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ 𝐵)
4	1, 2, 3	3eqtri 2848	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ 𝐵)

Syntax hints:   = wceq 1533   ∩ cin 3934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-in 3942

This theorem is referenced by:  in13  4198  inrot  4200  wefrc  5548  imainrect  6037  sspred  6155  fpwwe2  10064  incexclem  15190  setsfun  16517  setsfun0  16518  ressress  16561  kgeni  22144  kgencn3  22165  fclsrest  22631  voliunlem1  24150  bj-disj2r  34340  refrelsredund4  35866