Theorem iinin1 4154

Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4137 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
iinin1	|- ( A =/= (/) -> |^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem iinin1

Step	Hyp	Ref	Expression
1		iinin2 4153	. 2 |- ( A =/= (/) -> |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) )
2		incom 3525	. . . 4 |- ( C i^i B ) = ( B i^i C )
3	2	a1i 11	. . 3 |- ( x e. A -> ( C i^i B ) = ( B i^i C ) )
4	3	iineq2i 4104	. 2 |- |^|_ x e. A ( C i^i B ) = |^|_ x e. A ( B i^i C )
5		incom 3525	. 2 |- ( |^|_ x e. A C i^i B ) = ( B i^i |^|_ x e. A C )
6	1, 4, 5	3eqtr4g 2492	1 |- ( A =/= (/) -> |^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) )

Syntax hints:   -> wi 4   = wceq 1652   e. wcel 1725   =/= wne 2598   i^i cin 3311   (/)c0 3620   |^|_ciin 4086

This theorem is referenced by:  firest  13652

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-v 2950  df-dif 3315  df-in 3319  df-nul 3621  df-iin 4088