Theorem in13 4198

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 4197	. 2 ⊢ ((𝐵 ∩ 𝐶) ∩ 𝐴) = ((𝐵 ∩ 𝐴) ∩ 𝐶)
2		incom 4177	. 2 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐵 ∩ 𝐶) ∩ 𝐴)
3		incom 4177	. 2 ⊢ (𝐶 ∩ (𝐵 ∩ 𝐴)) = ((𝐵 ∩ 𝐴) ∩ 𝐶)
4	1, 2, 3	3eqtr4i 2854	1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴))

Syntax hints:   = wceq 1533   ∩ cin 3934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-in 3942

This theorem is referenced by:  inin  30271