Theorem in12 3467

Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)

Assertion

Ref	Expression
in12	⊢ (A ∩ (B ∩ C)) = (B ∩ (A ∩ C))

Proof of Theorem in12

Step	Hyp	Ref	Expression
1		incom 3449	. . 3 ⊢ (A ∩ B) = (B ∩ A)
2	1	ineq1i 3454	. 2 ⊢ ((A ∩ B) ∩ C) = ((B ∩ A) ∩ C)
3		inass 3466	. 2 ⊢ ((A ∩ B) ∩ C) = (A ∩ (B ∩ C))
4		inass 3466	. 2 ⊢ ((B ∩ A) ∩ C) = (B ∩ (A ∩ C))
5	2, 3, 4	3eqtr3i 2381	1 ⊢ (A ∩ (B ∩ C)) = (B ∩ (A ∩ C))

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:  in32  3468  in31  3470  in4  3472  resdmres  5079