Theorem difundir 3403

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 3399	. 2 |- ( ( A u. B ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ C ) ) u. ( B i^i ( _V \ C ) ) )
2		invdif 3392	. 2 |- ( ( A u. B ) i^i ( _V \ C ) ) = ( ( A u. B ) \ C )
3		invdif 3392	. . 3 |- ( A i^i ( _V \ C ) ) = ( A \ C )
4		invdif 3392	. . 3 |- ( B i^i ( _V \ C ) ) = ( B \ C )
5	3, 4	uneq12i 3302	. 2 |- ( ( A i^i ( _V \ C ) ) u. ( B i^i ( _V \ C ) ) ) = ( ( A \ C ) u. ( B \ C ) )
6	1, 2, 5	3eqtr3i 2218	1 |- ( ( A u. B ) \ C ) = ( ( A \ C ) u. ( B \ C ) )

Syntax hints:   = wceq 1364   _Vcvv 2752   \ cdif 3141   u. cun 3142   i^i cin 3143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-in 3150

This theorem is referenced by:  symdif1  3415  difun2  3517  diftpsn3  3748  unfiin  6943  setsfun0  12516  strleund  12581  strleun  12582