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Theorem abrexex2 5780
Description: Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x and  y. See also abrexex 5763. (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 1605 . . . 4  |-  F/ z E. x  e.  A  ph
2 nfcv 2419 . . . . 5  |-  F/_ y A
3 nfs1v 2045 . . . . 5  |-  F/ y [ z  /  y ] ph
42, 3nfrex 2598 . . . 4  |-  F/ y E. x  e.  A  [ z  /  y ] ph
5 sbequ12 1860 . . . . 5  |-  ( y  =  z  ->  ( ph 
<->  [ z  /  y ] ph ) )
65rexbidv 2564 . . . 4  |-  ( y  =  z  ->  ( E. x  e.  A  ph  <->  E. x  e.  A  [
z  /  y ]
ph ) )
71, 4, 6cbvab 2401 . . 3  |-  { y  |  E. x  e.  A  ph }  =  { z  |  E. x  e.  A  [
z  /  y ]
ph }
8 df-clab 2270 . . . . 5  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
98rexbii 2568 . . . 4  |-  ( E. x  e.  A  z  e.  { y  | 
ph }  <->  E. x  e.  A  [ z  /  y ] ph )
109abbii 2395 . . 3  |-  { z  |  E. x  e.  A  z  e.  {
y  |  ph } }  =  { z  |  E. x  e.  A  [ z  /  y ] ph }
117, 10eqtr4i 2306 . 2  |-  { y  |  E. x  e.  A  ph }  =  { z  |  E. x  e.  A  z  e.  { y  |  ph } }
12 df-iun 3907 . . 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 5770 . . 3  |-  U_ x  e.  A  { y  |  ph }  e.  _V
1612, 15eqeltrri 2354 . 2  |-  { z  |  E. x  e.  A  z  e.  {
y  |  ph } }  e.  _V
1711, 16eqeltri 2353 1  |-  { y  |  E. x  e.  A  ph }  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1623   [wsb 1629    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   U_ciun 3905
This theorem is referenced by:  abexssex  5781  abexex  5782  oprabrexex2  5963  ab2rexex  6000  ab2rexex2  6001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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