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Theorem abrexex2 6122
Description: Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x and  y. See also abrexex 6115. (Contributed by NM, 12-Sep-2004.)
Hypotheses
Ref Expression
abrexex2.1  |-  A  e. 
_V
abrexex2.2  |-  { y  |  ph }  e.  _V
Assertion
Ref Expression
abrexex2  |-  { y  |  E. x  e.  A  ph }  e.  _V
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem abrexex2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1528 . . . 4  |-  F/ z E. x  e.  A  ph
2 nfcv 2319 . . . . 5  |-  F/_ y A
3 nfs1v 1939 . . . . 5  |-  F/ y [ z  /  y ] ph
42, 3nfrexxy 2516 . . . 4  |-  F/ y E. x  e.  A  [ z  /  y ] ph
5 sbequ12 1771 . . . . 5  |-  ( y  =  z  ->  ( ph 
<->  [ z  /  y ] ph ) )
65rexbidv 2478 . . . 4  |-  ( y  =  z  ->  ( E. x  e.  A  ph  <->  E. x  e.  A  [
z  /  y ]
ph ) )
71, 4, 6cbvab 2301 . . 3  |-  { y  |  E. x  e.  A  ph }  =  { z  |  E. x  e.  A  [
z  /  y ]
ph }
8 df-clab 2164 . . . . 5  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
98rexbii 2484 . . . 4  |-  ( E. x  e.  A  z  e.  { y  | 
ph }  <->  E. x  e.  A  [ z  /  y ] ph )
109abbii 2293 . . 3  |-  { z  |  E. x  e.  A  z  e.  {
y  |  ph } }  =  { z  |  E. x  e.  A  [ z  /  y ] ph }
117, 10eqtr4i 2201 . 2  |-  { y  |  E. x  e.  A  ph }  =  { z  |  E. x  e.  A  z  e.  { y  |  ph } }
12 df-iun 3888 . . 3  |-  U_ x  e.  A  { y  |  ph }  =  {
z  |  E. x  e.  A  z  e.  { y  |  ph } }
13 abrexex2.1 . . . 4  |-  A  e. 
_V
14 abrexex2.2 . . . 4  |-  { y  |  ph }  e.  _V
1513, 14iunex 6121 . . 3  |-  U_ x  e.  A  { y  |  ph }  e.  _V
1612, 15eqeltrri 2251 . 2  |-  { z  |  E. x  e.  A  z  e.  {
y  |  ph } }  e.  _V
1711, 16eqeltri 2250 1  |-  { y  |  E. x  e.  A  ph }  e.  _V
Colors of variables: wff set class
Syntax hints:   [wsb 1762    e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2737   U_ciun 3886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223
This theorem is referenced by:  abexssex  6123  abexex  6124  oprabrexex2  6128  ab2rexex  6129  ab2rexex2  6130
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