Theorem elinti 1974

Description: Membership in class intersection.

Assertion

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

Proof of Theorem elinti

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 5 |- (x = A -> (x e. y <-> A e. y))
2	1	imbi2d 464	. . . 4 |- (x = A -> ((y e. B -> x e. y) <-> (y e. B -> A e. y)))
3	2	bialdv 935	. . 3 |- (x = A -> (A.y(y e. B -> x e. y) <-> A.y(y e. B -> A e. y)))
4		visset 1350	. . . . 5 |- x e. V
5	4	elint 1971	. . . 4 |- (x e. |^|B <-> A.y(y e. B -> x e. y))
6	5	biimp 133	. . 3 |- (x e. |^|B -> A.y(y e. B -> x e. y))
7	3, 6	vtoclga 1387	. 2 |- (A e. |^|B -> A.y(y e. B -> A e. y))
8		eleq1 1149	. . . . 5 |- (y = C -> (y e. B <-> C e. B))
9		eleq2 1150	. . . . 5 |- (y = C -> (A e. y <-> A e. C))
10	8, 9	imbi12d 474	. . . 4 |- (y = C -> ((y e. B -> A e. y) <-> (C e. B -> A e. C)))
11	10	cla4gv 1396	. . 3 |- (C e. B -> (A.y(y e. B -> A e. y) -> (C e. B -> A e. C)))
12	11	pm2.43b 61	. 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 2  A.wal 672   e. wel 803   = wceq 1091   e. wcel 1092  |^|cint 1965

This theorem is referenced by:  shintcl 5294

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-int 1966

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