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Theorem elintrabg 4241
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 2523 . 2  |-  ( y  =  A  ->  (
y  e.  |^| { x  e.  B  |  ph }  <->  A  e.  |^| { x  e.  B  |  ph }
) )
2 eleq1 2523 . . . 4  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32imbi2d 316 . . 3  |-  ( y  =  A  ->  (
( ph  ->  y  e.  x )  <->  ( ph  ->  A  e.  x ) ) )
43ralbidv 2838 . 2  |-  ( y  =  A  ->  ( A. x  e.  B  ( ph  ->  y  e.  x )  <->  A. x  e.  B  ( ph  ->  A  e.  x ) ) )
5 vex 3073 . . 3  |-  y  e. 
_V
65elintrab 4240 . 2  |-  ( y  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  y  e.  x
) )
71, 4, 6vtoclbg 3129 1  |-  ( A  e.  V  ->  ( A  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  e.  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   A.wral 2795   {crab 2799   |^|cint 4228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rab 2804  df-v 3072  df-int 4229
This theorem is referenced by:  tskmid  9110  eltskm  9113  nobndlem6  27974  elpcliN  33845
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