Theorem iunin1 4072

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4061 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 4071	. 2 |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )
2		incom 3421	. . . 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 4025	. 2 |- U_ x e. A ( C i^i B ) = U_ x e. A ( B i^i C )
5		incom 3421	. 2 |- ( U_ x e. A C i^i B ) = ( B i^i U_ x e. A C )
6	1, 4, 5	3eqtr4i 2269	1 |- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )

Syntax hints:   = wceq 1402   e. wcel 2209   i^i cin 3219   U_ciun 4007

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-iun 4009

This theorem is referenced by:  2iunin  4074  tgrest  15193