Theorem inrot 4203

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 4202	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴)
2		in32 4200	. 2 ⊢ ((𝐶 ∩ 𝐵) ∩ 𝐴) = ((𝐶 ∩ 𝐴) ∩ 𝐵)
3	1, 2	eqtri 2846	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵)

Syntax hints:   = wceq 1537   ∩ cin 3937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-in 3945

This theorem is referenced by: (None)