Theorem difindir 3192

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

Assertion

Ref	Expression
difindir	|- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) )

Proof of Theorem difindir

Step	Hyp	Ref	Expression
1		inindir 3155	. 2 |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ C ) ) i^i ( B i^i ( _V \ C ) ) )
2		invdif 3179	. 2 |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i B ) \ C )
3		invdif 3179	. . 3 |- ( A i^i ( _V \ C ) ) = ( A \ C )
4		invdif 3179	. . 3 |- ( B i^i ( _V \ C ) ) = ( B \ C )
5	3, 4	ineq12i 3136	. 2 |- ( ( A i^i ( _V \ C ) ) i^i ( B i^i ( _V \ C ) ) ) = ( ( A \ C ) i^i ( B \ C ) )
6	1, 2, 5	3eqtr3i 2068	1 |- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) )

Syntax hints:   = wceq 1243   _Vcvv 2557   \ cdif 2914   i^i cin 2916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924

This theorem is referenced by: (None)