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Theorem elintrabg 4006
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.
StepHypRef Expression
1 eleq1 2448 . 2  |-  ( y  =  A  ->  (
y  e.  |^| { x  e.  B  |  ph }  <->  A  e.  |^| { x  e.  B  |  ph }
) )
2 eleq1 2448 . . . 4  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32imbi2d 308 . . 3  |-  ( y  =  A  ->  (
( ph  ->  y  e.  x )  <->  ( ph  ->  A  e.  x ) ) )
43ralbidv 2670 . 2  |-  ( y  =  A  ->  ( A. x  e.  B  ( ph  ->  y  e.  x )  <->  A. x  e.  B  ( ph  ->  A  e.  x ) ) )
5 vex 2903 . . 3  |-  y  e. 
_V
65elintrab 4005 . 2  |-  ( y  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  y  e.  x
) )
71, 4, 6vtoclbg 2956 1  |-  ( A  e.  V  ->  ( A  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  e.  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   A.wral 2650   {crab 2654   |^|cint 3993
This theorem is referenced by:  tskmid  8649  eltskm  8652  nobndlem6  25376  elpcliN  30008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rab 2659  df-v 2902  df-int 3994
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