Theorem elintrabg 3628

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 2100	. 2 |- ( y = A -> ( y e. |^| { x e. B | ph } <-> A e. |^| { x e. B | ph } ) )
2		eleq1 2100	. . . 4 |- ( y = A -> ( y e. x <-> A e. x ) )
3	2	imbi2d 219	. . 3 |- ( y = A -> ( ( ph -> y e. x ) <-> ( ph -> A e. x ) ) )
4	3	ralbidv 2326	. 2 |- ( y = A -> ( A. x e. B ( ph -> y e. x ) <-> A. x e. B ( ph -> A e. x ) ) )
5		vex 2560	. . 3 |- y e. _V
6	5	elintrab 3627	. 2 |- ( y e. |^| { x e. B | ph } <-> A. x e. B ( ph -> y e. x ) )
7	1, 4, 6	vtoclbg 2614	1 |- ( A e. V -> ( A e. |^| { x e. B | ph } <-> A. x e. B ( ph -> A e. x ) ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   {crab 2310   |^|cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-int 3616

This theorem is referenced by: (None)