Theorem elinti 2542

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 1534	. . . . 5 |- (x = A -> (x e. y <-> A e. y))
2	1	imbi2d 612	. . . 4 |- (x = A -> ((y e. B -> x e. y) <-> (y e. B -> A e. y)))
3	2	albidv 1278	. . 3 |- (x = A -> (A.y(y e. B -> x e. y) <-> A.y(y e. B -> A e. y)))
4		visset 1813	. . . . 5 |- x e. V
5	4	elint 2539	. . . 4 |- (x e. |^|B <-> A.y(y e. B -> x e. y))
6	5	biimp 151	. . 3 |- (x e. |^|B -> A.y(y e. B -> x e. y))
7	3, 6	vtoclga 1852	. 2 |- (A e. |^|B -> A.y(y e. B -> A e. y))
8		eleq1 1534	. . . . 5 |- (y = C -> (y e. B <-> C e. B))
9		eleq2 1535	. . . . 5 |- (y = C -> (A e. y <-> A e. C))
10	8, 9	imbi12d 626	. . . 4 |- (y = C -> ((y e. B -> A e. y) <-> (C e. B -> A e. C)))
11	10	cla4gv 1862	. . 3 |- (C e. B -> (A.y(y e. B -> A e. y) -> (C e. B -> A e. C)))
12	11	pm2.43b 67	. 2 |- (A.y(y e. B -> A e. y) -> (C e. B -> A e. C))
13	7, 12	syl 10	1 |- (A e. |^|B -> (C e. B -> A e. C))

Syntax hints:   -> wi 3  A.wal 954   = wceq 956   e. wcel 958  |^|cint 2533

This theorem is referenced by:  shintcl 9293

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-int 2534

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