Theorem iunin2 4039

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4029 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 2691	. . . 4 |- ( E. x e. A ( y e. B /\ y e. C ) <-> ( y e. B /\ E. x e. A y e. C ) )
2		elin 3392	. . . . 5 |- ( y e. ( B i^i C ) <-> ( y e. B /\ y e. C ) )
3	2	rexbii 2540	. . . 4 |- ( E. x e. A y e. ( B i^i C ) <-> E. x e. A ( y e. B /\ y e. C ) )
4		eliun 3979	. . . . 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 3979	. . 3 |- ( y e. U_ x e. A ( B i^i C ) <-> E. x e. A y e. ( B i^i C ) )
8		elin 3392	. . 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 2228	1 |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2202   E.wrex 2512   i^i cin 3200   U_ciun 3975

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-in 3207  df-iun 3977

This theorem is referenced by:  iunin1  4040  2iunin  4042  resiun1  5038  resiun2  5039