MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abrexex2 Structured version   Visualization version   Unicode version

Theorem abrexex2 6761
Description: Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x and  y. See also abrexex 6754. (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 1764 . . . 4  |-  F/ z E. x  e.  A  ph
2 nfcv 2592 . . . . 5  |-  F/_ y A
3 nfs1v 2266 . . . . 5  |-  F/ y [ z  /  y ] ph
42, 3nfrex 2828 . . . 4  |-  F/ y E. x  e.  A  [ z  /  y ] ph
5 sbequ12 2083 . . . . 5  |-  ( y  =  z  ->  ( ph 
<->  [ z  /  y ] ph ) )
65rexbidv 2872 . . . 4  |-  ( y  =  z  ->  ( E. x  e.  A  ph  <->  E. x  e.  A  [
z  /  y ]
ph ) )
71, 4, 6cbvab 2574 . . 3  |-  { y  |  E. x  e.  A  ph }  =  { z  |  E. x  e.  A  [
z  /  y ]
ph }
8 df-clab 2438 . . . . 5  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
98rexbii 2861 . . . 4  |-  ( E. x  e.  A  z  e.  { y  | 
ph }  <->  E. x  e.  A  [ z  /  y ] ph )
109abbii 2567 . . 3  |-  { z  |  E. x  e.  A  z  e.  {
y  |  ph } }  =  { z  |  E. x  e.  A  [ z  /  y ] ph }
117, 10eqtr4i 2476 . 2  |-  { y  |  E. x  e.  A  ph }  =  { z  |  E. x  e.  A  z  e.  { y  |  ph } }
12 df-iun 4249 . . 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 6760 . . 3  |-  U_ x  e.  A  { y  |  ph }  e.  _V
1612, 15eqeltrri 2526 . 2  |-  { z  |  E. x  e.  A  z  e.  {
y  |  ph } }  e.  _V
1711, 16eqeltri 2525 1  |-  { y  |  E. x  e.  A  ph }  e.  _V
Colors of variables: wff setvar class
Syntax hints:   [wsb 1800    e. wcel 1890   {cab 2437   E.wrex 2737   _Vcvv 3012   U_ciun 4247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-rep 4486  ax-sep 4496  ax-nul 4505  ax-pr 4611  ax-un 6570
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-iun 4249  df-br 4374  df-opab 4433  df-mpt 4434  df-id 4726  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-f1 5565  df-fo 5566  df-f1o 5567  df-fv 5568
This theorem is referenced by:  abexssex  6762  abexex  6763  oprabrexex2  6770  ab2rexex  6771  ab2rexex2  6772
  Copyright terms: Public domain W3C validator