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Theorem moabex 4232
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 nfv 1605 . . 3  |-  F/ y
ph
21mo2 2172 . 2  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
3 abss 3242 . . . . 5  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  e. 
{ y } ) )
4 elsn 3655 . . . . . . 7  |-  ( x  e.  { y }  <-> 
x  =  y )
54imbi2i 303 . . . . . 6  |-  ( (
ph  ->  x  e.  {
y } )  <->  ( ph  ->  x  =  y ) )
65albii 1553 . . . . 5  |-  ( A. x ( ph  ->  x  e.  { y } )  <->  A. x ( ph  ->  x  =  y ) )
73, 6bitri 240 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  =  y ) )
8 snex 4216 . . . . 5  |-  { y }  e.  _V
98ssex 4158 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  ->  { x  |  ph }  e.  _V )
107, 9sylbir 204 . . 3  |-  ( A. x ( ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
1110exlimiv 1666 . 2  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
122, 11sylbi 187 1  |-  ( E* x ph  ->  { x  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   E*wmo 2144   {cab 2269   _Vcvv 2788    C_ wss 3152   {csn 3640
This theorem is referenced by:  rmorabex  4233  euabex  4234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647
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