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Theorem moabex 4651
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 2267 . 2  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
2 abss 3521 . . . . 5  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  e. 
{ y } ) )
3 elsn 3991 . . . . . . 7  |-  ( x  e.  { y }  <-> 
x  =  y )
43imbi2i 312 . . . . . 6  |-  ( (
ph  ->  x  e.  {
y } )  <->  ( ph  ->  x  =  y ) )
54albii 1611 . . . . 5  |-  ( A. x ( ph  ->  x  e.  { y } )  <->  A. x ( ph  ->  x  =  y ) )
62, 5bitri 249 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  =  y ) )
7 snex 4633 . . . . 5  |-  { y }  e.  _V
87ssex 4536 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  ->  { x  |  ph }  e.  _V )
96, 8sylbir 213 . . 3  |-  ( A. x ( ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
109exlimiv 1689 . 2  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
111, 10sylbi 195 1  |-  ( E* x ph  ->  { x  |  ph }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1368   E.wex 1587    e. wcel 1758   E*wmo 2261   {cab 2436   _Vcvv 3070    C_ wss 3428   {csn 3977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-sn 3978  df-pr 3980
This theorem is referenced by:  rmorabex  4652  euabex  4653
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