Theorem ssin 3684

Description: Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
ssin	|- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) )

Proof of Theorem ssin

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3649	. . . . 5 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
2	1	imbi2i 313	. . . 4 |- ( ( x e. A -> x e. ( B i^i C ) ) <-> ( x e. A -> ( x e. B /\ x e. C ) ) )
3	2	albii 1687	. . 3 |- ( A. x ( x e. A -> x e. ( B i^i C ) ) <-> A. x ( x e. A -> ( x e. B /\ x e. C ) ) )
4		jcab 871	. . . 4 |- ( ( x e. A -> ( x e. B /\ x e. C ) ) <-> ( ( x e. A -> x e. B ) /\ ( x e. A -> x e. C ) ) )
5	4	albii 1687	. . 3 |- ( A. x ( x e. A -> ( x e. B /\ x e. C ) ) <-> A. x ( ( x e. A -> x e. B ) /\ ( x e. A -> x e. C ) ) )
6		19.26 1725	. . 3 |- ( A. x ( ( x e. A -> x e. B ) /\ ( x e. A -> x e. C ) ) <-> ( A. x ( x e. A -> x e. B ) /\ A. x ( x e. A -> x e. C ) ) )
7	3, 5, 6	3bitrri 275	. 2 |- ( ( A. x ( x e. A -> x e. B ) /\ A. x ( x e. A -> x e. C ) ) <-> A. x ( x e. A -> x e. ( B i^i C ) ) )
8		dfss2 3453	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
9		dfss2 3453	. . 3 |- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
10	8, 9	anbi12i 701	. 2 |- ( ( A C_ B /\ A C_ C ) <-> ( A. x ( x e. A -> x e. B ) /\ A. x ( x e. A -> x e. C ) ) )
11		dfss2 3453	. 2 |- ( A C_ ( B i^i C ) <-> A. x ( x e. A -> x e. ( B i^i C ) ) )
12	7, 10, 11	3bitr4i 280	1 |- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   A.wal 1435   e. wcel 1868   i^i cin 3435   C_ wss 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-in 3443  df-ss 3450

This theorem is referenced by:  ssini  3685  ssind  3686  uneqin  3724  disjpss  3843  trin  4525  pwin  4753  fin  5776  wfrlem4  7043  epfrs  8216  tcmin  8226  resscntz  16972  subgdmdprd  17654  tgval  19956  eltg3i  19962  innei  20127  cnprest2  20292  subislly  20482  lly1stc  20497  xkohaus  20654  xkoinjcn  20688  opnfbas  20843  supfil  20896  rnelfm  20954  tsmsres  21144  restmetu  21571  chabs2  27155  cmbr4i  27239  pjin3i  27832  mdbr2  27934  dmdbr2  27941  dmdbr5  27946  mdslle1i  27955  mdslle2i  27956  mdslj1i  27957  mdslj2i  27958  mdsl2i  27960  mdslmd1lem1  27963  mdslmd1lem2  27964  mdslmd1i  27967  mdslmd3i  27970  hatomistici  28000  chrelat2i  28003  cvexchlem  28006  mdsymlem1  28041  mdsymlem3  28043  mdsymlem6  28046  dmdbr5ati  28060  pnfneige0  28752  ballotlem2  29316  iccllyscon  29968  frrlem4  30511  heibor1lem  32054  dochexmidlem1  34946  superficl  36090