Theorem iunin1 3953

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3942 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
iunin1	|- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )

Distinct variable group:   x, B

Allowed substitution hints:   A( x)   C( x)

Proof of Theorem iunin1

Step	Hyp	Ref	Expression
1		iunin2 3952	. 2 |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )
2		incom 3329	. . . 4 |- ( C i^i B ) = ( B i^i C )
3	2	a1i 9	. . 3 |- ( x e. A -> ( C i^i B ) = ( B i^i C ) )
4	3	iuneq2i 3906	. 2 |- U_ x e. A ( C i^i B ) = U_ x e. A ( B i^i C )
5		incom 3329	. 2 |- ( U_ x e. A C i^i B ) = ( B i^i U_ x e. A C )
6	1, 4, 5	3eqtr4i 2208	1 |- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )

Syntax hints:   = wceq 1353   e. wcel 2148   i^i cin 3130   U_ciun 3888

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-iun 3890

This theorem is referenced by:  2iunin  3955  tgrest  13754