Theorem in13 4156

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

Assertion

Ref	Expression
in13	⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴))

Proof of Theorem in13

Step	Hyp	Ref	Expression
1		in32 4155	. 2 ⊢ ((𝐵 ∩ 𝐶) ∩ 𝐴) = ((𝐵 ∩ 𝐴) ∩ 𝐶)
2		incom 4135	. 2 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐵 ∩ 𝐶) ∩ 𝐴)
3		incom 4135	. 2 ⊢ (𝐶 ∩ (𝐵 ∩ 𝐴)) = ((𝐵 ∩ 𝐴) ∩ 𝐶)
4	1, 2, 3	3eqtr4i 2776	1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴))

Syntax hints:   = wceq 1539   ∩ cin 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-in 3894

This theorem is referenced by:  inin  30862