Theorem iundif2 4033

Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4020 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 3221	. . . . 5 |- (y e. (B \ C) <-> (y e. B /\ -. y e. C))
2	1	rexbii 2639	. . . 4 |- (E.x e. A y e. (B \ C) <-> E.x e. A (y e. B /\ -. y e. C))
3		r19.42v 2765	. . . 4 |- (E.x e. A (y e. B /\ -. y e. C) <-> (y e. B /\ E.x e. A -. y e. C))
4		rexnal 2625	. . . . . 6 |- (E.x e. A -. y e. C <-> -. A.x e. A y e. C)
5		vex 2862	. . . . . . 7 |- y e. _V
6		eliin 3974	. . . . . . 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 302	. . . . 5 |- (E.x e. A -. y e. C <-> -. y e. |^|_x e. A C)
9	8	anbi2i 675	. . . 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 262	. . 3 |- (E.x e. A y e. (B \ C) <-> (y e. B /\ -. y e. |^|_x e. A C))
11		eliun 3973	. . 3 |- (y e. U_x e. A (B \ C) <-> E.x e. A y e. (B \ C))
12		eldif 3221	. . 3 |- (y e. (B \ |^|_x e. A C) <-> (y e. B /\ -. y e. |^|_x e. A C))
13	10, 11, 12	3bitr4i 268	. 2 |- (y e. U_x e. A (B \ C) <-> y e. (B \ |^|_x e. A C))
14	13	eqriv 2350	1 |- U_x e. A (B \ C) = (B \ |^|_x e. A C)

Syntax hints:  -. wn 3   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2614  E.wrex 2615  _Vcvv 2859   \ cdif 3206  U_ciun 3969  |^|_ciin 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-iun 3971  df-iin 3972

This theorem is referenced by: (None)