Theorem difundir 3509

Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
difundir	|- ((A u. B) \ C) = ((A \ C) u. (B \ C))

Proof of Theorem difundir

Step	Hyp	Ref	Expression
1		indir 3504	. 2 |- ((A u. B) i^i (_V \ C)) = ((A i^i (_V \ C)) u. (B i^i (_V \ C)))
2		invdif 3497	. 2 |- ((A u. B) i^i (_V \ C)) = ((A u. B) \ C)
3		invdif 3497	. . 3 |- (A i^i (_V \ C)) = (A \ C)
4		invdif 3497	. . 3 |- (B i^i (_V \ C)) = (B \ C)
5	3, 4	uneq12i 3417	. 2 |- ((A i^i (_V \ C)) u. (B i^i (_V \ C))) = ((A \ C) u. (B \ C))
6	1, 2, 5	3eqtr3i 2381	1 |- ((A u. B) \ C) = ((A \ C) u. (B \ C))

Syntax hints:   = wceq 1642  _Vcvv 2860   \ cdif 3207   u. cun 3208   i^i 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  df-un 3215  df-dif 3216

This theorem is referenced by:  symdif1  3520  difun2  3630  diftpsn3  3850