Theorem iinin1 4591

Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4574 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 4590	. 2 |- ( A =/= (/) -> |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) )
2		incom 3805	. . . 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 4540	. 2 |- |^|_ x e. A ( C i^i B ) = |^|_ x e. A ( B i^i C )
5		incom 3805	. 2 |- ( |^|_ x e. A C i^i B ) = ( B i^i |^|_ x e. A C )
6	1, 4, 5	3eqtr4g 2681	1 |- ( A =/= (/) -> |^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   (/)c0 3915   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-iin 4523

This theorem is referenced by:  firest  16093  iniin1  39309