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Theorem elintab 4134
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 4129 . 2  |-  ( A  e.  |^| { x  | 
ph }  <->  A. y
( y  e.  {
x  |  ph }  ->  A  e.  y ) )
3 nfsab1 2428 . . . 4  |-  F/ x  y  e.  { x  |  ph }
4 nfv 1673 . . . 4  |-  F/ x  A  e.  y
53, 4nfim 1852 . . 3  |-  F/ x
( y  e.  {
x  |  ph }  ->  A  e.  y )
6 nfv 1673 . . 3  |-  F/ y ( ph  ->  A  e.  x )
7 eleq1 2498 . . . . 5  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
8 abid 2426 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
97, 8syl6bb 261 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  ph ) )
10 eleq2 2499 . . . 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 1969 . 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 1367    e. wcel 1756   {cab 2424   _Vcvv 2967   |^|cint 4123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2969  df-int 4124
This theorem is referenced by:  elintrab  4135  intmin4  4152  intab  4153  intid  4545  dfom3  7845  dfom5  7848  tc2  7954  dfnn2  10327  efgi  16207  efgi2  16213  heibor1lem  28661
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