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Theorem elintab 4240
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
inteqab.1  |-  A  e. 
_V
Assertion
Ref Expression
elintab  |-  ( A  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  e.  x ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem elintab
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 inteqab.1 . . 3  |-  A  e. 
_V
21elint 4235 . 2  |-  ( A  e.  |^| { x  | 
ph }  <->  A. y
( y  e.  {
x  |  ph }  ->  A  e.  y ) )
3 nfsab1 2440 . . . 4  |-  F/ x  y  e.  { x  |  ph }
4 nfv 1674 . . . 4  |-  F/ x  A  e.  y
53, 4nfim 1855 . . 3  |-  F/ x
( y  e.  {
x  |  ph }  ->  A  e.  y )
6 nfv 1674 . . 3  |-  F/ y ( ph  ->  A  e.  x )
7 eleq1 2523 . . . . 5  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
8 abid 2438 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
97, 8syl6bb 261 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  ph ) )
10 eleq2 2524 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
119, 10imbi12d 320 . . 3  |-  ( y  =  x  ->  (
( y  e.  {
x  |  ph }  ->  A  e.  y )  <-> 
( ph  ->  A  e.  x ) ) )
125, 6, 11cbval 1978 . 2  |-  ( A. y ( y  e. 
{ x  |  ph }  ->  A  e.  y )  <->  A. x ( ph  ->  A  e.  x ) )
132, 12bitri 249 1  |-  ( A  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  e.  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    e. wcel 1758   {cab 2436   _Vcvv 3071   |^|cint 4229
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-v 3073  df-int 4230
This theorem is referenced by:  elintrab  4241  intmin4  4258  intab  4259  intid  4651  dfom3  7957  dfom5  7960  tc2  8066  dfnn2  10439  efgi  16329  efgi2  16335  heibor1lem  28849
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