Theorem ssin 3355

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 3316	. . . . 5 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
2	1	imbi2i 226	. . . 4 |- ( ( x e. A -> x e. ( B i^i C ) ) <-> ( x e. A -> ( x e. B /\ x e. C ) ) )
3	2	albii 1468	. . 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 603	. . . 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 1468	. . 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 1479	. . 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 207	. 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 3142	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
9		dfss2 3142	. . 3 |- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
10	8, 9	anbi12i 460	. 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 3142	. 2 |- ( A C_ ( B i^i C ) <-> A. x ( x e. A -> x e. ( B i^i C ) ) )
12	7, 10, 11	3bitr4i 212	1 |- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   e. wcel 2146   i^i cin 3126   C_ wss 3127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-in 3133  df-ss 3140

This theorem is referenced by:  ssini  3356  ssind  3357  uneqin  3384  trin  4106  pwin  4276  peano5  4591  fin  5394  tgval  13100  eltg3i  13107  innei  13214  cnptoprest2  13291