Theorem in31 4183

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 4180	. 2 ⊢ (𝐶 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ (𝐶 ∩ 𝐵))
2		incom 4161	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∩ 𝐵))
3		incom 4161	. 2 ⊢ ((𝐶 ∩ 𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐶 ∩ 𝐵))
4	1, 2, 3	3eqtr4i 2853	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴)

Syntax hints:   = wceq 1537   ∩ cin 3918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3142  df-v 3483  df-in 3926

This theorem is referenced by:  inrot  4184