Theorem iinun2 2606

Description: Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 2599 to recover Enderton's theorem.

Assertion

Ref	Expression
iinun2	|- |^|_x e. A (B u. C) = (B u. |^|_x e. A C)

Distinct variable group:   x,B

Proof of Theorem iinun2

Step	Hyp	Ref	Expression
1		r19.32v 1757	. . . 4 |- (A.x e. A (y e. B \/ y e. C) <-> (y e. B \/ A.x e. A y e. C))
2		elun 2171	. . . . 5 |- (y e. (B u. C) <-> (y e. B \/ y e. C))
3	2	ralbii 1666	. . . 4 |- (A.x e. A y e. (B u. C) <-> A.x e. A (y e. B \/ y e. C))
4		visset 1811	. . . . . 6 |- y e. V
5		eliin 2568	. . . . . 6 |- (y e. V -> (y e. |^|_x e. A C <-> A.x e. A y e. C))
6	4, 5	ax-mp 7	. . . . 5 |- (y e. |^|_x e. A C <-> A.x e. A y e. C)
7	6	orbi2i 255	. . . 4 |- ((y e. B \/ y e. |^|_x e. A C) <-> (y e. B \/ A.x e. A y e. C))
8	1, 3, 7	3bitr4 183	. . 3 |- (A.x e. A y e. (B u. C) <-> (y e. B \/ y e. |^|_x e. A C))
9		eliin 2568	. . . 4 |- (y e. V -> (y e. |^|_x e. A (B u. C) <-> A.x e. A y e. (B u. C)))
10	4, 9	ax-mp 7	. . 3 |- (y e. |^|_x e. A (B u. C) <-> A.x e. A y e. (B u. C))
11		elun 2171	. . 3 |- (y e. (B u. |^|_x e. A C) <-> (y e. B \/ y e. |^|_x e. A C))
12	8, 10, 11	3bitr4 183	. 2 |- (y e. |^|_x e. A (B u. C) <-> y e. (B u. |^|_x e. A C))
13	12	eqriv 1474	1 |- |^|_x e. A (B u. C) = (B u. |^|_x e. A C)

Syntax hints:   <-> wb 146   \/ wo 222   = wceq 955   e. wcel 957  A.wral 1644  Vcvv 1809   u. cun 2043  |^|_ciin 2564

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1648  df-v 1810  df-un 2048  df-iin 2566

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