Theorem elintrabg 2546

Description: Membership in the intersection of a class abstraction.

Assertion

Ref	Expression
elintrabg	|- (A e. C -> (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x)))

Distinct variable group:   x,A

Proof of Theorem elintrabg

Step	Hyp	Ref	Expression
1		eleq1 1534	. 2 |- (y = A -> (y e. |^|{x e. B | ph} <-> A e. |^|{x e. B | ph}))
2		eleq1 1534	. . . 4 |- (y = A -> (y e. x <-> A e. x))
3	2	imbi2d 612	. . 3 |- (y = A -> ((ph -> y e. x) <-> (ph -> A e. x)))
4	3	ralbidv 1663	. 2 |- (y = A -> (A.x e. B (ph -> y e. x) <-> A.x e. B (ph -> A e. x)))
5		visset 1813	. . 3 |- y e. V
6	5	elintrab 2545	. 2 |- (y e. |^|{x e. B | ph} <-> A.x e. B (ph -> y e. x))
7	1, 4, 6	vtoclbg 1848	1 |- (A e. C -> (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x)))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648  |^|cint 2533

This theorem is referenced by:  islp2 7747

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-rab 1652  df-v 1812  df-int 2534

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