Theorem in13 3788

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 3787	. 2 ⊢ ((𝐵 ∩ 𝐶) ∩ 𝐴) = ((𝐵 ∩ 𝐴) ∩ 𝐶)
2		incom 3767	. 2 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐵 ∩ 𝐶) ∩ 𝐴)
3		incom 3767	. 2 ⊢ (𝐶 ∩ (𝐵 ∩ 𝐴)) = ((𝐵 ∩ 𝐴) ∩ 𝐶)
4	1, 2, 3	3eqtr4i 2642	1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴))

Syntax hints:   = wceq 1475   ∩ cin 3539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547

This theorem is referenced by:  inin  28737