Theorem elintg 2541

Description: Membership in class intersection, with the sethood requirement expressed as an antecedent.

Assertion

Ref	Expression
elintg	|- (A e. C -> (A e. |^|B <-> A.x e. B A e. x))

Distinct variable groups:   x,A   x,B

Proof of Theorem elintg

Step	Hyp	Ref	Expression
1		eleq1 1534	. . 3 |- (y = A -> (y e. |^|B <-> A e. |^|B))
2		eleq1 1534	. . . 4 |- (y = A -> (y e. x <-> A e. x))
3	2	ralbidv 1663	. . 3 |- (y = A -> (A.x e. B y e. x <-> A.x e. B A e. x))
4	1, 3	bibi12d 629	. 2 |- (y = A -> ((y e. |^|B <-> A.x e. B y e. x) <-> (A e. |^|B <-> A.x e. B A e. x)))
5		visset 1813	. . 3 |- y e. V
6	5	elint2 2540	. 2 |- (y e. |^|B <-> A.x e. B y e. x)
7	4, 6	vtoclg 1847	1 |- (A e. C -> (A e. |^|B <-> A.x e. B A e. x))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  A.wral 1645  |^|cint 2533

This theorem is referenced by:  onmindif 3060  onmindif2 3061

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-ral 1649  df-v 1812  df-int 2534

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