Theorem inindi 3473

Description: Intersection distributes over itself. (Contributed by NM, 6-May-1994.)

Assertion

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

Proof of Theorem inindi

Step	Hyp	Ref	Expression
1		inidm 3465	. . 3 ⊢ (A ∩ A) = A
2	1	ineq1i 3454	. 2 ⊢ ((A ∩ A) ∩ (B ∩ C)) = (A ∩ (B ∩ C))
3		in4 3472	. 2 ⊢ ((A ∩ A) ∩ (B ∩ C)) = ((A ∩ B) ∩ (A ∩ C))
4	2, 3	eqtr3i 2375	1 ⊢ (A ∩ (B ∩ C)) = ((A ∩ 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:  difundi  3508  dfif5  3675  inindif  4076  resindi  4984