Theorem difundir 3380

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 3376	. 2 |- ( ( A u. B ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ C ) ) u. ( B i^i ( _V \ C ) ) )
2		invdif 3369	. 2 |- ( ( A u. B ) i^i ( _V \ C ) ) = ( ( A u. B ) \ C )
3		invdif 3369	. . 3 |- ( A i^i ( _V \ C ) ) = ( A \ C )
4		invdif 3369	. . 3 |- ( B i^i ( _V \ C ) ) = ( B \ C )
5	3, 4	uneq12i 3279	. 2 |- ( ( A i^i ( _V \ C ) ) u. ( B i^i ( _V \ C ) ) ) = ( ( A \ C ) u. ( B \ C ) )
6	1, 2, 5	3eqtr3i 2199	1 |- ( ( A u. B ) \ C ) = ( ( A \ C ) u. ( B \ C ) )

Syntax hints:   = wceq 1348   _Vcvv 2730   \ cdif 3118   u. cun 3119   i^i cin 3120

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127

This theorem is referenced by:  symdif1  3392  difun2  3494  diftpsn3  3721  unfiin  6903  setsfun0  12452  strleund  12506  strleun  12507