Theorem iindif2 4036

Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4020 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)

Assertion

Ref	Expression
iindif2	|- (A =/= (/) -> |^|_x e. A (B \ C) = (B \ U_x e. A C))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   C(x)

Proof of Theorem iindif2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.28zv 3646	. . . 4 |- (A =/= (/) -> (A.x e. A (y e. B /\ -. y e. C) <-> (y e. B /\ A.x e. A -. y e. C)))
2		eldif 3222	. . . . . 6 |- (y e. (B \ C) <-> (y e. B /\ -. y e. C))
3	2	bicomi 193	. . . . 5 |- ((y e. B /\ -. y e. C) <-> y e. (B \ C))
4	3	ralbii 2639	. . . 4 |- (A.x e. A (y e. B /\ -. y e. C) <-> A.x e. A y e. (B \ C))
5		ralnex 2625	. . . . . 6 |- (A.x e. A -. y e. C <-> -. E.x e. A y e. C)
6		eliun 3974	. . . . . 6 |- (y e. U_x e. A C <-> E.x e. A y e. C)
7	5, 6	xchbinxr 302	. . . . 5 |- (A.x e. A -. y e. C <-> -. y e. U_x e. A C)
8	7	anbi2i 675	. . . 4 |- ((y e. B /\ A.x e. A -. y e. C) <-> (y e. B /\ -. y e. U_x e. A C))
9	1, 4, 8	3bitr3g 278	. . 3 |- (A =/= (/) -> (A.x e. A y e. (B \ C) <-> (y e. B /\ -. y e. U_x e. A C)))
10		vex 2863	. . . 4 |- y e. _V
11		eliin 3975	. . . 4 |- (y e. _V -> (y e. |^|_x e. A (B \ C) <-> A.x e. A y e. (B \ C)))
12	10, 11	ax-mp 5	. . 3 |- (y e. |^|_x e. A (B \ C) <-> A.x e. A y e. (B \ C))
13		eldif 3222	. . 3 |- (y e. (B \ U_x e. A C) <-> (y e. B /\ -. y e. U_x e. A C))
14	9, 12, 13	3bitr4g 279	. 2 |- (A =/= (/) -> (y e. |^|_x e. A (B \ C) <-> y e. (B \ U_x e. A C)))
15	14	eqrdv 2351	1 |- (A =/= (/) -> |^|_x e. A (B \ C) = (B \ U_x e. A C))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  A.wral 2615  E.wrex 2616  _Vcvv 2860   \ cdif 3207  (/)c0 3551  U_ciun 3970  |^|_ciin 3971

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-iun 3972  df-iin 3973

This theorem is referenced by: (None)