Theorem ssin 3478

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 3220	. . . . 5 |- (x e. (B i^i C) <-> (x e. B /\ x e. C))
2	1	imbi2i 303	. . . 4 |- ((x e. A -> x e. (B i^i C)) <-> (x e. A -> (x e. B /\ x e. C)))
3	2	albii 1566	. . 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 833	. . . 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 1566	. . 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 1593	. . 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 263	. 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 3263	. . 3 |- (A C_ B <-> A.x(x e. A -> x e. B))
9		dfss2 3263	. . 3 |- (A C_ C <-> A.x(x e. A -> x e. C))
10	8, 9	anbi12i 678	. 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 3263	. 2 |- (A C_ (B i^i C) <-> A.x(x e. A -> x e. (B i^i C)))
12	7, 10, 11	3bitr4i 268	1 |- ((A C_ B /\ A C_ C) <-> A C_ (B i^i C))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   e. wcel 1710   i^i cin 3209   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  ssini  3479  ssind  3480  uneqin  3507  disjpss  3602  fin  5247  clos1induct  5881  sbthlem1  6204