Theorem undir 3459

Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Assertion

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

Proof of Theorem undir

Step	Hyp	Ref	Expression
1		undi 3457	. 2 |- ( C u. ( A i^i B ) ) = ( ( C u. A ) i^i ( C u. B ) )
2		uncom 3353	. 2 |- ( ( A i^i B ) u. C ) = ( C u. ( A i^i B ) )
3		uncom 3353	. . 3 |- ( A u. C ) = ( C u. A )
4		uncom 3353	. . 3 |- ( B u. C ) = ( C u. B )
5	3, 4	ineq12i 3408	. 2 |- ( ( A u. C ) i^i ( B u. C ) ) = ( ( C u. A ) i^i ( C u. B ) )
6	1, 2, 5	3eqtr4i 2262	1 |- ( ( A i^i B ) u. C ) = ( ( A u. C ) i^i ( B u. C ) )

Syntax hints:   = wceq 1398   u. cun 3199   i^i cin 3200

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207

This theorem is referenced by: (None)