Theorem in31 3215

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

Assertion

Ref	Expression
in31	⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴)

Proof of Theorem in31

Step	Hyp	Ref	Expression
1		in12 3212	. 2 ⊢ (𝐶 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ (𝐶 ∩ 𝐵))
2		incom 3193	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∩ 𝐵))
3		incom 3193	. 2 ⊢ ((𝐶 ∩ 𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐶 ∩ 𝐵))
4	1, 2, 3	3eqtr4i 2119	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴)

Syntax hints:   = wceq 1290   ∩ cin 2999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006

This theorem is referenced by:  inrot  3216