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Theorem elintrabg 4288
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 2532 . 2  |-  ( y  =  A  ->  (
y  e.  |^| { x  e.  B  |  ph }  <->  A  e.  |^| { x  e.  B  |  ph }
) )
2 eleq1 2532 . . . 4  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32imbi2d 316 . . 3  |-  ( y  =  A  ->  (
( ph  ->  y  e.  x )  <->  ( ph  ->  A  e.  x ) ) )
43ralbidv 2896 . 2  |-  ( y  =  A  ->  ( A. x  e.  B  ( ph  ->  y  e.  x )  <->  A. x  e.  B  ( ph  ->  A  e.  x ) ) )
5 vex 3109 . . 3  |-  y  e. 
_V
65elintrab 4287 . 2  |-  ( y  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  y  e.  x
) )
71, 4, 6vtoclbg 3165 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 1374    e. wcel 1762   A.wral 2807   {crab 2811   |^|cint 4275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rab 2816  df-v 3108  df-int 4276
This theorem is referenced by:  tskmid  9207  eltskm  9210  nobndlem6  29020  elpcliN  34564
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