Theorem iunin2 4028

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4018 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
iunin2	|- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )

Distinct variable group:   x, B

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

Proof of Theorem iunin2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.42v 2688	. . . 4 |- ( E. x e. A ( y e. B /\ y e. C ) <-> ( y e. B /\ E. x e. A y e. C ) )
2		elin 3387	. . . . 5 |- ( y e. ( B i^i C ) <-> ( y e. B /\ y e. C ) )
3	2	rexbii 2537	. . . 4 |- ( E. x e. A y e. ( B i^i C ) <-> E. x e. A ( y e. B /\ y e. C ) )
4		eliun 3968	. . . . 5 |- ( y e. U_ x e. A C <-> E. x e. A y e. C )
5	4	anbi2i 457	. . . 4 |- ( ( y e. B /\ y e. U_ x e. A C ) <-> ( y e. B /\ E. x e. A y e. C ) )
6	1, 3, 5	3bitr4i 212	. . 3 |- ( E. x e. A y e. ( B i^i C ) <-> ( y e. B /\ y e. U_ x e. A C ) )
7		eliun 3968	. . 3 |- ( y e. U_ x e. A ( B i^i C ) <-> E. x e. A y e. ( B i^i C ) )
8		elin 3387	. . 3 |- ( y e. ( B i^i U_ x e. A C ) <-> ( y e. B /\ y e. U_ x e. A C ) )
9	6, 7, 8	3bitr4i 212	. 2 |- ( y e. U_ x e. A ( B i^i C ) <-> y e. ( B i^i U_ x e. A C ) )
10	9	eqriv 2226	1 |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   E.wrex 2509   i^i cin 3196   U_ciun 3964

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-iun 3966

This theorem is referenced by:  iunin1  4029  2iunin  4031  resiun1  5023  resiun2  5024