Theorem inrot 4198

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 4197	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴)
2		in32 4195	. 2 ⊢ ((𝐶 ∩ 𝐵) ∩ 𝐴) = ((𝐶 ∩ 𝐴) ∩ 𝐵)
3	1, 2	eqtri 2753	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵)

Syntax hints:   = wceq 1540   ∩ cin 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-in 3923

This theorem is referenced by: (None)