Theorem iunin1 4006

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3995 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
iunin1	|- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )

Distinct variable group:   x, B

Allowed substitution hints:   A( x)   C( x)

Proof of Theorem iunin1

Step	Hyp	Ref	Expression
1		iunin2 4005	. 2 |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )
2		incom 3373	. . . 4 |- ( C i^i B ) = ( B i^i C )
3	2	a1i 9	. . 3 |- ( x e. A -> ( C i^i B ) = ( B i^i C ) )
4	3	iuneq2i 3959	. 2 |- U_ x e. A ( C i^i B ) = U_ x e. A ( B i^i C )
5		incom 3373	. 2 |- ( U_ x e. A C i^i B ) = ( B i^i U_ x e. A C )
6	1, 4, 5	3eqtr4i 2238	1 |- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )

Syntax hints:   = wceq 1373   e. wcel 2178   i^i cin 3173   U_ciun 3941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-iun 3943

This theorem is referenced by:  2iunin  4008  tgrest  14756