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Theorem abrexex2 6003
Description: Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x and  y. See also abrexex 5985. (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 1630 . . . 4  |-  F/ z E. x  e.  A  ph
2 nfcv 2574 . . . . 5  |-  F/_ y A
3 nfs1v 2184 . . . . 5  |-  F/ y [ z  /  y ] ph
42, 3nfrex 2763 . . . 4  |-  F/ y E. x  e.  A  [ z  /  y ] ph
5 sbequ12 1945 . . . . 5  |-  ( y  =  z  ->  ( ph 
<->  [ z  /  y ] ph ) )
65rexbidv 2728 . . . 4  |-  ( y  =  z  ->  ( E. x  e.  A  ph  <->  E. x  e.  A  [
z  /  y ]
ph ) )
71, 4, 6cbvab 2556 . . 3  |-  { y  |  E. x  e.  A  ph }  =  { z  |  E. x  e.  A  [
z  /  y ]
ph }
8 df-clab 2425 . . . . 5  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
98rexbii 2732 . . . 4  |-  ( E. x  e.  A  z  e.  { y  | 
ph }  <->  E. x  e.  A  [ z  /  y ] ph )
109abbii 2550 . . 3  |-  { z  |  E. x  e.  A  z  e.  {
y  |  ph } }  =  { z  |  E. x  e.  A  [ z  /  y ] ph }
117, 10eqtr4i 2461 . 2  |-  { y  |  E. x  e.  A  ph }  =  { z  |  E. x  e.  A  z  e.  { y  |  ph } }
12 df-iun 4097 . . 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 5993 . . 3  |-  U_ x  e.  A  { y  |  ph }  e.  _V
1612, 15eqeltrri 2509 . 2  |-  { z  |  E. x  e.  A  z  e.  {
y  |  ph } }  e.  _V
1711, 16eqeltri 2508 1  |-  { y  |  E. x  e.  A  ph }  e.  _V
Colors of variables: wff set class
Syntax hints:   [wsb 1659    e. wcel 1726   {cab 2424   E.wrex 2708   _Vcvv 2958   U_ciun 4095
This theorem is referenced by:  abexssex  6004  abexex  6005  oprabrexex2  6191  ab2rexex  6228  ab2rexex2  6229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464
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