Theorem in13 4183

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 4182	. 2 ⊢ ((𝐵 ∩ 𝐶) ∩ 𝐴) = ((𝐵 ∩ 𝐴) ∩ 𝐶)
2		incom 4162	. 2 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐵 ∩ 𝐶) ∩ 𝐴)
3		incom 4162	. 2 ⊢ (𝐶 ∩ (𝐵 ∩ 𝐴)) = ((𝐵 ∩ 𝐴) ∩ 𝐶)
4	1, 2, 3	3eqtr4i 2798	1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴))

This proof depends on syntax axioms:   = wceq 1570   ∩ cin 3905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-in 3913

This theorem is used by:  inin  32935