Theorem iinin2 4590

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
iinin2	|- ( A =/= (/) -> |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem iinin2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.28zv 4066	. . . 4 |- ( A =/= (/) -> ( A. x e. A ( y e. B /\ y e. C ) <-> ( y e. B /\ A. x e. A y e. C ) ) )
2		elin 3796	. . . . 5 |- ( y e. ( B i^i C ) <-> ( y e. B /\ y e. C ) )
3	2	ralbii 2980	. . . 4 |- ( A. x e. A y e. ( B i^i C ) <-> A. x e. A ( y e. B /\ y e. C ) )
4		vex 3203	. . . . . 6 |- y e. _V
5		eliin 4525	. . . . . 6 |- ( y e. _V -> ( y e. |^|_ x e. A C <-> A. x e. A y e. C ) )
6	4, 5	ax-mp 5	. . . . 5 |- ( y e. |^|_ x e. A C <-> A. x e. A y e. C )
7	6	anbi2i 730	. . . 4 |- ( ( y e. B /\ y e. |^|_ x e. A C ) <-> ( y e. B /\ A. x e. A y e. C ) )
8	1, 3, 7	3bitr4g 303	. . 3 |- ( A =/= (/) -> ( A. x e. A y e. ( B i^i C ) <-> ( y e. B /\ y e. |^|_ x e. A C ) ) )
9		eliin 4525	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. A ( B i^i C ) <-> A. x e. A y e. ( B i^i C ) ) )
10	4, 9	ax-mp 5	. . 3 |- ( y e. |^|_ x e. A ( B i^i C ) <-> A. x e. A y e. ( B i^i C ) )
11		elin 3796	. . 3 |- ( y e. ( B i^i |^|_ x e. A C ) <-> ( y e. B /\ y e. |^|_ x e. A C ) )
12	8, 10, 11	3bitr4g 303	. 2 |- ( A =/= (/) -> ( y e. |^|_ x e. A ( B i^i C ) <-> y e. ( B i^i |^|_ x e. A C ) ) )
13	12	eqrdv 2620	1 |- ( A =/= (/) -> |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   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:  iinin1  4591  iniin2  39310