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Theorem elintab 3840
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 3835 . 2  |-  ( A  e.  |^| { x  | 
ph }  <->  A. y
( y  e.  {
x  |  ph }  ->  A  e.  y ) )
3 nfsab1 2160 . . . 4  |-  F/ x  y  e.  { x  |  ph }
4 nfv 1521 . . . 4  |-  F/ x  A  e.  y
53, 4nfim 1565 . . 3  |-  F/ x
( y  e.  {
x  |  ph }  ->  A  e.  y )
6 nfv 1521 . . 3  |-  F/ y ( ph  ->  A  e.  x )
7 eleq1 2233 . . . . 5  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
8 abid 2158 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
97, 8bitrdi 195 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  ph ) )
10 eleq2 2234 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
119, 10imbi12d 233 . . 3  |-  ( y  =  x  ->  (
( y  e.  {
x  |  ph }  ->  A  e.  y )  <-> 
( ph  ->  A  e.  x ) ) )
125, 6, 11cbval 1747 . 2  |-  ( A. y ( y  e. 
{ x  |  ph }  ->  A  e.  y )  <->  A. x ( ph  ->  A  e.  x ) )
132, 12bitri 183 1  |-  ( A  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1346    e. wcel 2141   {cab 2156   _Vcvv 2730   |^|cint 3829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-int 3830
This theorem is referenced by:  elintrab  3841  intmin4  3857  intab  3858  intid  4207
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