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Theorem abrexex2 6765
Description: Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x and  y. See also abrexex 6758. (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 1683 . . . 4  |-  F/ z E. x  e.  A  ph
2 nfcv 2629 . . . . 5  |-  F/_ y A
3 nfs1v 2164 . . . . 5  |-  F/ y [ z  /  y ] ph
42, 3nfrex 2927 . . . 4  |-  F/ y E. x  e.  A  [ z  /  y ] ph
5 sbequ12 1961 . . . . 5  |-  ( y  =  z  ->  ( ph 
<->  [ z  /  y ] ph ) )
65rexbidv 2973 . . . 4  |-  ( y  =  z  ->  ( E. x  e.  A  ph  <->  E. x  e.  A  [
z  /  y ]
ph ) )
71, 4, 6cbvab 2608 . . 3  |-  { y  |  E. x  e.  A  ph }  =  { z  |  E. x  e.  A  [
z  /  y ]
ph }
8 df-clab 2453 . . . . 5  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
98rexbii 2965 . . . 4  |-  ( E. x  e.  A  z  e.  { y  | 
ph }  <->  E. x  e.  A  [ z  /  y ] ph )
109abbii 2601 . . 3  |-  { z  |  E. x  e.  A  z  e.  {
y  |  ph } }  =  { z  |  E. x  e.  A  [ z  /  y ] ph }
117, 10eqtr4i 2499 . 2  |-  { y  |  E. x  e.  A  ph }  =  { z  |  E. x  e.  A  z  e.  { y  |  ph } }
12 df-iun 4327 . . 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 6764 . . 3  |-  U_ x  e.  A  { y  |  ph }  e.  _V
1612, 15eqeltrri 2552 . 2  |-  { z  |  E. x  e.  A  z  e.  {
y  |  ph } }  e.  _V
1711, 16eqeltri 2551 1  |-  { y  |  E. x  e.  A  ph }  e.  _V
Colors of variables: wff setvar class
Syntax hints:   [wsb 1711    e. wcel 1767   {cab 2452   E.wrex 2815   _Vcvv 3113   U_ciun 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595
This theorem is referenced by:  abexssex  6766  abexex  6767  oprabrexex2  6774  ab2rexex  6775  ab2rexex2  6776
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