Theorem difundir 3459

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 3455	. 2 |- ( ( A u. B ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ C ) ) u. ( B i^i ( _V \ C ) ) )
2		invdif 3448	. 2 |- ( ( A u. B ) i^i ( _V \ C ) ) = ( ( A u. B ) \ C )
3		invdif 3448	. . 3 |- ( A i^i ( _V \ C ) ) = ( A \ C )
4		invdif 3448	. . 3 |- ( B i^i ( _V \ C ) ) = ( B \ C )
5	3, 4	uneq12i 3358	. 2 |- ( ( A i^i ( _V \ C ) ) u. ( B i^i ( _V \ C ) ) ) = ( ( A \ C ) u. ( B \ C ) )
6	1, 2, 5	3eqtr3i 2259	1 |- ( ( A u. B ) \ C ) = ( ( A \ C ) u. ( B \ C ) )

Syntax hints:   = wceq 1397   _Vcvv 2801   \ cdif 3196   u. cun 3197   i^i cin 3198

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-dif 3201  df-un 3203  df-in 3205

This theorem is referenced by:  symdif1  3471  difun2  3573  diftpsn3  3815  unfiin  7123  setsfun0  13141  strleund  13209  strleun  13210