Theorem difundir 3329

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 3325	. 2 |- ( ( A u. B ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ C ) ) u. ( B i^i ( _V \ C ) ) )
2		invdif 3318	. 2 |- ( ( A u. B ) i^i ( _V \ C ) ) = ( ( A u. B ) \ C )
3		invdif 3318	. . 3 |- ( A i^i ( _V \ C ) ) = ( A \ C )
4		invdif 3318	. . 3 |- ( B i^i ( _V \ C ) ) = ( B \ C )
5	3, 4	uneq12i 3228	. 2 |- ( ( A i^i ( _V \ C ) ) u. ( B i^i ( _V \ C ) ) ) = ( ( A \ C ) u. ( B \ C ) )
6	1, 2, 5	3eqtr3i 2168	1 |- ( ( A u. B ) \ C ) = ( ( A \ C ) u. ( B \ C ) )

Syntax hints:   = wceq 1331   _Vcvv 2686   \ cdif 3068   u. cun 3069   i^i cin 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077

This theorem is referenced by:  symdif1  3341  difun2  3442  diftpsn3  3661  unfiin  6814  setsfun0  11995  strleund  12047  strleun  12048