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Theorem elintab 4299
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 4294 . 2  |-  ( A  e.  |^| { x  | 
ph }  <->  A. y
( y  e.  {
x  |  ph }  ->  A  e.  y ) )
3 nfsab1 2446 . . . 4  |-  F/ x  y  e.  { x  |  ph }
4 nfv 1708 . . . 4  |-  F/ x  A  e.  y
53, 4nfim 1921 . . 3  |-  F/ x
( y  e.  {
x  |  ph }  ->  A  e.  y )
6 nfv 1708 . . 3  |-  F/ y ( ph  ->  A  e.  x )
7 eleq1 2529 . . . . 5  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
8 abid 2444 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
97, 8syl6bb 261 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  ph ) )
10 eleq2 2530 . . . 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 2022 . 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 1393    e. wcel 1819   {cab 2442   _Vcvv 3109   |^|cint 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-int 4289
This theorem is referenced by:  elintrab  4300  intmin4  4318  intab  4319  intid  4714  dfom3  8081  dfom5  8084  tc2  8190  dfnn2  10569  efgi  16864  efgi2  16870  mclsax  29126  heibor1lem  30510  brintclab  37923  lem1frege76a  38157
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