Theorem ssin 3293

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 3254	. . . . 5 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
2	1	imbi2i 225	. . . 4 |- ( ( x e. A -> x e. ( B i^i C ) ) <-> ( x e. A -> ( x e. B /\ x e. C ) ) )
3	2	albii 1446	. . 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 592	. . . 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 1446	. . 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 1457	. . 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 206	. 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 3081	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
9		dfss2 3081	. . 3 |- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
10	8, 9	anbi12i 455	. 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 3081	. 2 |- ( A C_ ( B i^i C ) <-> A. x ( x e. A -> x e. ( B i^i C ) ) )
12	7, 10, 11	3bitr4i 211	1 |- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329   e. wcel 1480   i^i cin 3065   C_ wss 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079

This theorem is referenced by:  ssini  3294  ssind  3295  uneqin  3322  trin  4031  pwin  4199  peano5  4507  fin  5304  tgval  12207  eltg3i  12214  innei  12321  cnptoprest2  12398