Theorem iundif2 2615

Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 2606 to recover Enderton's theorem.

Assertion

Ref	Expression
iundif2	|- U_x e. A (B \ C) = (B \ |^|_x e. A C)

Distinct variable group:   x,B

Proof of Theorem iundif2

Step	Hyp	Ref	Expression
1		r19.42v 1767	. . . 4 |- (E.x e. A (y e. B /\ -. y e. C) <-> (y e. B /\ E.x e. A -. y e. C))
2		eldif 2060	. . . . 5 |- (y e. (B \ C) <-> (y e. B /\ -. y e. C))
3	2	rexbii 1671	. . . 4 |- (E.x e. A y e. (B \ C) <-> E.x e. A (y e. B /\ -. y e. C))
4		visset 1816	. . . . . . . 8 |- y e. V
5		eliin 2575	. . . . . . . 8 |- (y e. V -> (y e. |^|_x e. A C <-> A.x e. A y e. C))
6	4, 5	ax-mp 7	. . . . . . 7 |- (y e. |^|_x e. A C <-> A.x e. A y e. C)
7	6	negbii 187	. . . . . 6 |- (-. y e. |^|_x e. A C <-> -. A.x e. A y e. C)
8		rexnal 1657	. . . . . 6 |- (E.x e. A -. y e. C <-> -. A.x e. A y e. C)
9	7, 8	bitr4 176	. . . . 5 |- (-. y e. |^|_x e. A C <-> E.x e. A -. y e. C)
10	9	anbi2i 482	. . . 4 |- ((y e. B /\ -. y e. |^|_x e. A C) <-> (y e. B /\ E.x e. A -. y e. C))
11	1, 3, 10	3bitr4 183	. . 3 |- (E.x e. A y e. (B \ C) <-> (y e. B /\ -. y e. |^|_x e. A C))
12		eliun 2574	. . 3 |- (y e. U_x e. A (B \ C) <-> E.x e. A y e. (B \ C))
13		eldif 2060	. . 3 |- (y e. (B \ |^|_x e. A C) <-> (y e. B /\ -. y e. |^|_x e. A C))
14	11, 12, 13	3bitr4 183	. 2 |- (y e. U_x e. A (B \ C) <-> y e. (B \ |^|_x e. A C))
15	14	eqriv 1477	1 |- U_x e. A (B \ C) = (B \ |^|_x e. A C)

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649  Vcvv 1814   \ cdif 2047  U_ciun 2570  |^|_ciin 2571

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-iun 2572  df-iin 2573

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