Theorem iinin2 4037

Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4021 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 3646	. . . 4 |- (A =/= (/) -> (A.x e. A (y e. B /\ y e. C) <-> (y e. B /\ A.x e. A y e. C)))
2		elin 3220	. . . . 5 |- (y e. (B i^i C) <-> (y e. B /\ y e. C))
3	2	ralbii 2639	. . . 4 |- (A.x e. A y e. (B i^i C) <-> A.x e. A (y e. B /\ y e. C))
4		vex 2863	. . . . . 6 |- y e. _V
5		eliin 3975	. . . . . 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 675	. . . 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 279	. . 3 |- (A =/= (/) -> (A.x e. A y e. (B i^i C) <-> (y e. B /\ y e. |^|_x e. A C)))
9		eliin 3975	. . . 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 3220	. . 3 |- (y e. (B i^i |^|_x e. A C) <-> (y e. B /\ y e. |^|_x e. A C))
12	8, 10, 11	3bitr4g 279	. 2 |- (A =/= (/) -> (y e. |^|_x e. A (B i^i C) <-> y e. (B i^i |^|_x e. A C)))
13	12	eqrdv 2351	1 |- (A =/= (/) -> |^|_x e. A (B i^i C) = (B i^i |^|_x e. A C))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  A.wral 2615  _Vcvv 2860   i^i cin 3209  (/)c0 3551  |^|_ciin 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-iin 3973

This theorem is referenced by:  iinin1  4038