Theorem iinin2m 3716

Description: Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)

Assertion

Ref	Expression
iinin2m	|- (E.x x e. 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 iinin2m

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.28mv 3308	. . . 4 |- (E.x x e. A -> (A.x e. A (y e. B /\ y e. C) <-> (y e. B /\ A.x e. A y e. C)))
2		elin 3120	. . . . 5 |- (y e. (B i^i C) <-> (y e. B /\ y e. C))
3	2	ralbii 2324	. . . 4 |- (A.x e. A y e. (B i^i C) <-> A.x e. A (y e. B /\ y e. C))
4		vex 2554	. . . . . 6 |- y e. _V
5		eliin 3653	. . . . . 6 |- (y e. _V -> (y e. |^|_x e. A C <-> A.x e. A y e. C))
6	4, 5	ax-mp 7	. . . . 5 |- (y e. |^|_x e. A C <-> A.x e. A y e. C)
7	6	anbi2i 430	. . . 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 212	. . 3 |- (E.x x e. A -> (A.x e. A y e. (B i^i C) <-> (y e. B /\ y e. |^|_x e. A C)))
9		eliin 3653	. . . 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 7	. . 3 |- (y e. |^|_x e. A (B i^i C) <-> A.x e. A y e. (B i^i C))
11		elin 3120	. . 3 |- (y e. (B i^i |^|_x e. A C) <-> (y e. B /\ y e. |^|_x e. A C))
12	8, 10, 11	3bitr4g 212	. 2 |- (E.x x e. A -> (y e. |^|_x e. A (B i^i C) <-> y e. (B i^i |^|_x e. A C)))
13	12	eqrdv 2035	1 |- (E.x x e. A -> |^|_x e. A (B i^i C) = (B i^i |^|_x e. A C))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242  E.wex 1378   e. wcel 1390  A.wral 2300   _Vcvv 2551   i^i cin 2910   |^|_ciin 3649

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-iin 3651

This theorem is referenced by:  iinin1m  3717