Theorem iundif2 4587

Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4574 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Assertion

Ref	Expression
iundif2	|- U_ x e. A ( B \ C ) = ( B \ |^|_ x e. A C )

Distinct variable group:   x, B

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

Proof of Theorem iundif2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3584	. . . . 5 |- ( y e. ( B \ C ) <-> ( y e. B /\ -. y e. C ) )
2	1	rexbii 3041	. . . 4 |- ( E. x e. A y e. ( B \ C ) <-> E. x e. A ( y e. B /\ -. y e. C ) )
3		r19.42v 3092	. . . 4 |- ( E. x e. A ( y e. B /\ -. y e. C ) <-> ( y e. B /\ E. x e. A -. y e. C ) )
4		rexnal 2995	. . . . . 6 |- ( E. x e. A -. y e. C <-> -. A. x e. A y e. C )
5		vex 3203	. . . . . . 7 |- y e. _V
6		eliin 4525	. . . . . . 7 |- ( y e. _V -> ( y e. |^|_ x e. A C <-> A. x e. A y e. C ) )
7	5, 6	ax-mp 5	. . . . . 6 |- ( y e. |^|_ x e. A C <-> A. x e. A y e. C )
8	4, 7	xchbinxr 325	. . . . 5 |- ( E. x e. A -. y e. C <-> -. y e. |^|_ x e. A C )
9	8	anbi2i 730	. . . 4 |- ( ( y e. B /\ E. x e. A -. y e. C ) <-> ( y e. B /\ -. y e. |^|_ x e. A C ) )
10	2, 3, 9	3bitri 286	. . 3 |- ( E. x e. A y e. ( B \ C ) <-> ( y e. B /\ -. y e. |^|_ x e. A C ) )
11		eliun 4524	. . 3 |- ( y e. U_ x e. A ( B \ C ) <-> E. x e. A y e. ( B \ C ) )
12		eldif 3584	. . 3 |- ( y e. ( B \ |^|_ x e. A C ) <-> ( y e. B /\ -. y e. |^|_ x e. A C ) )
13	10, 11, 12	3bitr4i 292	. 2 |- ( y e. U_ x e. A ( B \ C ) <-> y e. ( B \ |^|_ x e. A C ) )
14	13	eqriv 2619	1 |- U_ x e. A ( B \ C ) = ( B \ |^|_ x e. A C )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   U_ciun 4520   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-iun 4522  df-iin 4523

This theorem is referenced by:  iuncld  20849  pnrmopn  21147  alexsublem  21848  bcth3  23128  iundifdifd  29380  iundifdif  29381