Theorem iunin2 3949

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3939 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 2634	. . . 4 |- ( E. x e. A ( y e. B /\ y e. C ) <-> ( y e. B /\ E. x e. A y e. C ) )
2		elin 3318	. . . . 5 |- ( y e. ( B i^i C ) <-> ( y e. B /\ y e. C ) )
3	2	rexbii 2484	. . . 4 |- ( E. x e. A y e. ( B i^i C ) <-> E. x e. A ( y e. B /\ y e. C ) )
4		eliun 3890	. . . . 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 3890	. . 3 |- ( y e. U_ x e. A ( B i^i C ) <-> E. x e. A y e. ( B i^i C ) )
8		elin 3318	. . 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 2174	1 |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2148   E.wrex 2456   i^i cin 3128   U_ciun 3886

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 2739  df-in 3135  df-iun 3888

This theorem is referenced by:  iunin1  3950  2iunin  3952  resiun1  4925  resiun2  4926