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Theorem elintrabg 3891
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 2356 . 2  |-  ( y  =  A  ->  (
y  e.  |^| { x  e.  B  |  ph }  <->  A  e.  |^| { x  e.  B  |  ph }
) )
2 eleq1 2356 . . . 4  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32imbi2d 307 . . 3  |-  ( y  =  A  ->  (
( ph  ->  y  e.  x )  <->  ( ph  ->  A  e.  x ) ) )
43ralbidv 2576 . 2  |-  ( y  =  A  ->  ( A. x  e.  B  ( ph  ->  y  e.  x )  <->  A. x  e.  B  ( ph  ->  A  e.  x ) ) )
5 vex 2804 . . 3  |-  y  e. 
_V
65elintrab 3890 . 2  |-  ( y  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  y  e.  x
) )
71, 4, 6vtoclbg 2857 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 176    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   |^|cint 3878
This theorem is referenced by:  tskmid  8478  eltskm  8481  nobndlem6  24422  intopcoaconc  25644  elpcliN  30704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-int 3879
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