Theorem ssin 3339

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 3300	. . . . 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 1457	. . 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 593	. . . 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 1457	. . 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 1468	. . 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 3126	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
9		dfss2 3126	. . 3 |- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
10	8, 9	anbi12i 456	. 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 3126	. 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 1340   e. wcel 2135   i^i cin 3110   C_ wss 3111

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-in 3117  df-ss 3124

This theorem is referenced by:  ssini  3340  ssind  3341  uneqin  3368  trin  4084  pwin  4254  peano5  4569  fin  5368  tgval  12590  eltg3i  12597  innei  12704  cnptoprest2  12781