Theorem elintiOLD 3224

Description: Membership in class intersection.

Assertion

Ref	Expression
elintiOLD	|- (A e. |^|B -> (C e. B -> A e. C))

Proof of Theorem elintiOLD

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . 5 |- (x = A -> (x e. y <-> A e. y))
2	1	imbi2d 674	. . . 4 |- (x = A -> ((y e. B -> x e. y) <-> (y e. B -> A e. y)))
3	2	albidv 1656	. . 3 |- (x = A -> (A.y(y e. B -> x e. y) <-> A.y(y e. B -> A e. y)))
4		visset 2295	. . . . 5 |- x e. _V
5	4	elint 3220	. . . 4 |- (x e. |^|B <-> A.y(y e. B -> x e. y))
6	5	biimpi 168	. . 3 |- (x e. |^|B -> A.y(y e. B -> x e. y))
7	3, 6	vtoclga 2352	. 2 |- (A e. |^|B -> A.y(y e. B -> A e. y))
8		eleq1 1957	. . . . 5 |- (y = C -> (y e. B <-> C e. B))
9		eleq2 1958	. . . . 5 |- (y = C -> (A e. y <-> A e. C))
10	8, 9	imbi12d 688	. . . 4 |- (y = C -> ((y e. B -> A e. y) <-> (C e. B -> A e. C)))
11	10	cla4gv 2364	. . 3 |- (C e. B -> (A.y(y e. B -> A e. y) -> (C e. B -> A e. C)))
12	11	pm2.43b 81	. 2 |- (A.y(y e. B -> A e. y) -> (C e. B -> A e. C))
13	7, 12	syl 12	1 |- (A e. |^|B -> (C e. B -> A e. C))

Syntax hints:   -> wi 3  A.wal 1296   = wceq 1298   e. wcel 1300  |^|cint 3214

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-int 3215

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