Theorem iunin2 3980

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3970 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 2654	. . . 4 |- ( E. x e. A ( y e. B /\ y e. C ) <-> ( y e. B /\ E. x e. A y e. C ) )
2		elin 3346	. . . . 5 |- ( y e. ( B i^i C ) <-> ( y e. B /\ y e. C ) )
3	2	rexbii 2504	. . . 4 |- ( E. x e. A y e. ( B i^i C ) <-> E. x e. A ( y e. B /\ y e. C ) )
4		eliun 3920	. . . . 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 3920	. . 3 |- ( y e. U_ x e. A ( B i^i C ) <-> E. x e. A y e. ( B i^i C ) )
8		elin 3346	. . 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 2193	1 |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2167   E.wrex 2476   i^i cin 3156   U_ciun 3916

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-iun 3918

This theorem is referenced by:  iunin1  3981  2iunin  3983  resiun1  4965  resiun2  4966