Theorem iunin2 3936

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3926 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 2627	. . . 4 |- ( E. x e. A ( y e. B /\ y e. C ) <-> ( y e. B /\ E. x e. A y e. C ) )
2		elin 3310	. . . . 5 |- ( y e. ( B i^i C ) <-> ( y e. B /\ y e. C ) )
3	2	rexbii 2477	. . . 4 |- ( E. x e. A y e. ( B i^i C ) <-> E. x e. A ( y e. B /\ y e. C ) )
4		eliun 3877	. . . . 5 |- ( y e. U_ x e. A C <-> E. x e. A y e. C )
5	4	anbi2i 454	. . . 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 211	. . 3 |- ( E. x e. A y e. ( B i^i C ) <-> ( y e. B /\ y e. U_ x e. A C ) )
7		eliun 3877	. . 3 |- ( y e. U_ x e. A ( B i^i C ) <-> E. x e. A y e. ( B i^i C ) )
8		elin 3310	. . 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 211	. 2 |- ( y e. U_ x e. A ( B i^i C ) <-> y e. ( B i^i U_ x e. A C ) )
10	9	eqriv 2167	1 |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )

Syntax hints:   /\ wa 103   = wceq 1348   e. wcel 2141   E.wrex 2449   i^i cin 3120   U_ciun 3873

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-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-iun 3875

This theorem is referenced by:  iunin1  3937  2iunin  3939  resiun1  4910  resiun2  4911