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Theorem iinab 4152
Description: Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
Assertion
Ref Expression
iinab  |-  |^|_ x  e.  A  { y  |  ph }  =  {
y  |  A. x  e.  A  ph }
Distinct variable groups:    y, A    x, y
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem iinab
StepHypRef Expression
1 nfcv 2572 . . . 4  |-  F/_ y A
2 nfab1 2574 . . . 4  |-  F/_ y { y  |  ph }
31, 2nfiin 4120 . . 3  |-  F/_ y |^|_ x  e.  A  {
y  |  ph }
4 nfab1 2574 . . 3  |-  F/_ y { y  |  A. x  e.  A  ph }
53, 4cleqf 2596 . 2  |-  ( |^|_ x  e.  A  { y  |  ph }  =  { y  |  A. x  e.  A  ph }  <->  A. y ( y  e. 
|^|_ x  e.  A  { y  |  ph } 
<->  y  e.  { y  |  A. x  e.  A  ph } ) )
6 abid 2424 . . . 4  |-  ( y  e.  { y  | 
ph }  <->  ph )
76ralbii 2729 . . 3  |-  ( A. x  e.  A  y  e.  { y  |  ph } 
<-> 
A. x  e.  A  ph )
8 vex 2959 . . . 4  |-  y  e. 
_V
9 eliin 4098 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  { y  |  ph }  <->  A. x  e.  A  y  e.  { y  |  ph }
) )
108, 9ax-mp 8 . . 3  |-  ( y  e.  |^|_ x  e.  A  { y  |  ph } 
<-> 
A. x  e.  A  y  e.  { y  |  ph } )
11 abid 2424 . . 3  |-  ( y  e.  { y  | 
A. x  e.  A  ph }  <->  A. x  e.  A  ph )
127, 10, 113bitr4i 269 . 2  |-  ( y  e.  |^|_ x  e.  A  { y  |  ph } 
<->  y  e.  { y  |  A. x  e.  A  ph } )
135, 12mpgbir 1559 1  |-  |^|_ x  e.  A  { y  |  ph }  =  {
y  |  A. x  e.  A  ph }
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705   _Vcvv 2956   |^|_ciin 4094
This theorem is referenced by:  iinrab  4153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-iin 4096
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