Theorem in31 4197

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 4194	. 2 ⊢ (𝐶 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ (𝐶 ∩ 𝐵))
2		incom 4174	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∩ 𝐵))
3		incom 4174	. 2 ⊢ ((𝐶 ∩ 𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐶 ∩ 𝐵))
4	1, 2, 3	3eqtr4i 2763	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴)

Syntax hints:   = wceq 1540   ∩ cin 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-in 3923

This theorem is referenced by:  inrot  4198