Theorem in31 3470

Description: A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)

Assertion

Ref	Expression
in31	⊢ ((A ∩ B) ∩ C) = ((C ∩ B) ∩ A)

Proof of Theorem in31

Step	Hyp	Ref	Expression
1		in12 3467	. 2 ⊢ (C ∩ (A ∩ B)) = (A ∩ (C ∩ B))
2		incom 3449	. 2 ⊢ ((A ∩ B) ∩ C) = (C ∩ (A ∩ B))
3		incom 3449	. 2 ⊢ ((C ∩ B) ∩ A) = (A ∩ (C ∩ B))
4	1, 2, 3	3eqtr4i 2383	1 ⊢ ((A ∩ B) ∩ C) = ((C ∩ B) ∩ A)

Syntax hints:   = wceq 1642   ∩ cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214

This theorem is referenced by:  inrot  3471