Theorem 2iunin 3323

Description: Rearrange indexed unions over intersection.

Assertion

Ref	Expression
2iunin	|- U_x e. A U_y e. B (C i^i D) = (U_x e. A C i^i U_y e. B D)

Distinct variable groups:   x,B   y,C   x,D   x,y

Proof of Theorem 2iunin

Step	Hyp	Ref	Expression
1		iunin2 3320	. . . 4 |- U_y e. B (C i^i D) = (C i^i U_y e. B D)
2	1	a1i 8	. . 3 |- (x e. A -> U_y e. B (C i^i D) = (C i^i U_y e. B D))
3	2	iuneq2i 3276	. 2 |- U_x e. A U_y e. B (C i^i D) = U_x e. A (C i^i U_y e. B D)
4		incom 2787	. . . . 5 |- (C i^i U_y e. B D) = (U_y e. B D i^i C)
5	4	a1i 8	. . . 4 |- (x e. A -> (C i^i U_y e. B D) = (U_y e. B D i^i C))
6	5	iuneq2i 3276	. . 3 |- U_x e. A (C i^i U_y e. B D) = U_x e. A (U_y e. B D i^i C)
7		iunin2 3320	. . 3 |- U_x e. A (U_y e. B D i^i C) = (U_y e. B D i^i U_x e. A C)
8	6, 7	eqtri 1908	. 2 |- U_x e. A (C i^i U_y e. B D) = (U_y e. B D i^i U_x e. A C)
9		incom 2787	. 2 |- (U_y e. B D i^i U_x e. A C) = (U_x e. A C i^i U_y e. B D)
10	3, 8, 9	3eqtri 1912	1 |- U_x e. A U_y e. B (C i^i D) = (U_x e. A C i^i U_y e. B D)

Syntax hints:   = wceq 1298   e. wcel 1300   i^i cin 2592  U_ciun 3255

This theorem is referenced by:  fpar 5085

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-in 2603  df-ss 2605  df-iun 3257

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