Theorem elintrabg 3748

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)   V( x)

Proof of Theorem elintrabg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2175	. 2 |- ( y = A -> ( y e. |^| { x e. B | ph } <-> A e. |^| { x e. B | ph } ) )
2		eleq1 2175	. . . 4 |- ( y = A -> ( y e. x <-> A e. x ) )
3	2	imbi2d 229	. . 3 |- ( y = A -> ( ( ph -> y e. x ) <-> ( ph -> A e. x ) ) )
4	3	ralbidv 2409	. 2 |- ( y = A -> ( A. x e. B ( ph -> y e. x ) <-> A. x e. B ( ph -> A e. x ) ) )
5		vex 2658	. . 3 |- y e. _V
6	5	elintrab 3747	. 2 |- ( y e. |^| { x e. B | ph } <-> A. x e. B ( ph -> y e. x ) )
7	1, 4, 6	vtoclbg 2716	1 |- ( A e. V -> ( A e. |^| { x e. B | ph } <-> A. x e. B ( ph -> A e. x ) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1312   e. wcel 1461   A.wral 2388   {crab 2392   |^|cint 3735

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rab 2397  df-v 2657  df-int 3736

This theorem is referenced by: (None)