Theorem undir 3377

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 3375	. 2 |- ( C u. ( A i^i B ) ) = ( ( C u. A ) i^i ( C u. B ) )
2		uncom 3271	. 2 |- ( ( A i^i B ) u. C ) = ( C u. ( A i^i B ) )
3		uncom 3271	. . 3 |- ( A u. C ) = ( C u. A )
4		uncom 3271	. . 3 |- ( B u. C ) = ( C u. B )
5	3, 4	ineq12i 3326	. 2 |- ( ( A u. C ) i^i ( B u. C ) ) = ( ( C u. A ) i^i ( C u. B ) )
6	1, 2, 5	3eqtr4i 2201	1 |- ( ( A i^i B ) u. C ) = ( ( A u. C ) i^i ( B u. C ) )

Syntax hints:   = wceq 1348   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-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-un 3125  df-in 3127

This theorem is referenced by: (None)