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Theorem moabex 4390
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 1626 . . 3  |-  F/ y
ph
21mo2 2291 . 2  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
3 abss 3380 . . . . 5  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  e. 
{ y } ) )
4 elsn 3797 . . . . . . 7  |-  ( x  e.  { y }  <-> 
x  =  y )
54imbi2i 304 . . . . . 6  |-  ( (
ph  ->  x  e.  {
y } )  <->  ( ph  ->  x  =  y ) )
65albii 1572 . . . . 5  |-  ( A. x ( ph  ->  x  e.  { y } )  <->  A. x ( ph  ->  x  =  y ) )
73, 6bitri 241 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  =  y ) )
8 snex 4373 . . . . 5  |-  { y }  e.  _V
98ssex 4315 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  ->  { x  |  ph }  e.  _V )
107, 9sylbir 205 . . 3  |-  ( A. x ( ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
1110exlimiv 1641 . 2  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
122, 11sylbi 188 1  |-  ( E* x ph  ->  { x  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546   E.wex 1547    e. wcel 1721   E*wmo 2263   {cab 2398   _Vcvv 2924    C_ wss 3288   {csn 3782
This theorem is referenced by:  rmorabex  4391  euabex  4392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-sn 3788  df-pr 3789
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