Theorem in32 3359

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 3357	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶))
2		in12 3358	. 2 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶))
3		incom 3339	. 2 ⊢ (𝐵 ∩ (𝐴 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ 𝐵)
4	1, 2, 3	3eqtri 2212	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ 𝐵)

Syntax hints:   = wceq 1363   ∩ cin 3140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-in 3147

This theorem is referenced by:  in13  3360  inrot  3362  imainrect  5086  setsfun  12511  setsfun0  12512