Theorem indir 3186

Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
indir	|- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) )

Proof of Theorem indir

Step	Hyp	Ref	Expression
1		indi 3184	. 2 |- ( C i^i ( A u. B ) ) = ( ( C i^i A ) u. ( C i^i B ) )
2		incom 3129	. 2 |- ( ( A u. B ) i^i C ) = ( C i^i ( A u. B ) )
3		incom 3129	. . 3 |- ( A i^i C ) = ( C i^i A )
4		incom 3129	. . 3 |- ( B i^i C ) = ( C i^i B )
5	3, 4	uneq12i 3095	. 2 |- ( ( A i^i C ) u. ( B i^i C ) ) = ( ( C i^i A ) u. ( C i^i B ) )
6	1, 2, 5	3eqtr4i 2070	1 |- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) )

Syntax hints:   = wceq 1243   u. cun 2915   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-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-un 2922  df-in 2924

This theorem is referenced by:  difundir  3190  undisj1  3279  disjpr2  3434  resundir  4626