Theorem elintrabg 3784

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 2202	. 2 |- ( y = A -> ( y e. |^| { x e. B | ph } <-> A e. |^| { x e. B | ph } ) )
2		eleq1 2202	. . . 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 2437	. 2 |- ( y = A -> ( A. x e. B ( ph -> y e. x ) <-> A. x e. B ( ph -> A e. x ) ) )
5		vex 2689	. . 3 |- y e. _V
6	5	elintrab 3783	. 2 |- ( y e. |^| { x e. B | ph } <-> A. x e. B ( ph -> y e. x ) )
7	1, 4, 6	vtoclbg 2747	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 1331   e. wcel 1480   A.wral 2416   {crab 2420   |^|cint 3771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-int 3772

This theorem is referenced by: (None)