Theorem 2iunin 4364

Description: Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)

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

Allowed substitution hints:   A( x, y)   B( y)   C( x)   D( y)

Proof of Theorem 2iunin

Step	Hyp	Ref	Expression
1		iunin2 4360	. . . 4 |- U_ y e. B ( C i^i D ) = ( C i^i U_ y e. B D )
2	1	a1i 11	. . 3 |- ( x e. A -> U_ y e. B ( C i^i D ) = ( C i^i U_ y e. B D ) )
3	2	iuneq2i 4315	. 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		iunin1 4361	. 2 |- U_ x e. A ( C i^i U_ y e. B D ) = ( U_ x e. A C i^i U_ y e. B D )
5	3, 4	eqtri 2451	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 1437   e. wcel 1868   i^i cin 3435   U_ciun 4296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rex 2781  df-v 3083  df-in 3443  df-ss 3450  df-iun 4298

This theorem is referenced by:  fpar  6908