Theorem inrot 4217

Description: Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)

Assertion

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

Proof of Theorem inrot

Step	Hyp	Ref	Expression
1		in31 4216	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴)
2		in32 4214	. 2 ⊢ ((𝐶 ∩ 𝐵) ∩ 𝐴) = ((𝐶 ∩ 𝐴) ∩ 𝐵)
3	1, 2	eqtri 2752	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵)

Syntax hints:   = wceq 1533   ∩ cin 3940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-in 3948

This theorem is referenced by: (None)