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Theorem iinab 3979
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 2432 . . . 4  |-  F/_ y A
2 nfab1 2434 . . . 4  |-  F/_ y { y  |  ph }
31, 2nfiin 3948 . . 3  |-  F/_ y |^|_ x  e.  A  {
y  |  ph }
4 nfab1 2434 . . 3  |-  F/_ y { y  |  A. x  e.  A  ph }
53, 4cleqf 2456 . 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 2284 . . . 4  |-  ( y  e.  { y  | 
ph }  <->  ph )
76ralbii 2580 . . 3  |-  ( A. x  e.  A  y  e.  { y  |  ph } 
<-> 
A. x  e.  A  ph )
8 vex 2804 . . . 4  |-  y  e. 
_V
9 eliin 3926 . . . 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 2284 . . 3  |-  ( y  e.  { y  | 
A. x  e.  A  ph }  <->  A. x  e.  A  ph )
127, 10, 113bitr4i 268 . 2  |-  ( y  e.  |^|_ x  e.  A  { y  |  ph } 
<->  y  e.  { y  |  A. x  e.  A  ph } )
135, 12mpgbir 1540 1  |-  |^|_ x  e.  A  { y  |  ph }  =  {
y  |  A. x  e.  A  ph }
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   _Vcvv 2801   |^|_ciin 3922
This theorem is referenced by:  iinrab  3980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-iin 3924
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