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Theorem moabex 4632
Description: "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)
Assertion
Ref Expression
moabex  |-  ( E* x ph  ->  { x  |  ph }  e.  _V )

Proof of Theorem moabex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 mo2v 2307 . 2  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
2 abss 3466 . . . . 5  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  e. 
{ y } ) )
3 elsn 3950 . . . . . . 7  |-  ( x  e.  { y }  <-> 
x  =  y )
43imbi2i 318 . . . . . 6  |-  ( (
ph  ->  x  e.  {
y } )  <->  ( ph  ->  x  =  y ) )
54albii 1695 . . . . 5  |-  ( A. x ( ph  ->  x  e.  { y } )  <->  A. x ( ph  ->  x  =  y ) )
62, 5bitri 257 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  =  y ) )
7 snex 4614 . . . . 5  |-  { y }  e.  _V
87ssex 4519 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  ->  { x  |  ph }  e.  _V )
96, 8sylbir 218 . . 3  |-  ( A. x ( ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
109exlimiv 1780 . 2  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
111, 10sylbi 200 1  |-  ( E* x ph  ->  { x  |  ph }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1446   E.wex 1667    e. wcel 1891   E*wmo 2301   {cab 2438   _Vcvv 3013    C_ wss 3372   {csn 3936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pr 4612
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-v 3015  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-sn 3937  df-pr 3939
This theorem is referenced by:  rmorabex  4633  euabex  4634
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