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Theorem moabex 4424
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 1630 . . 3  |-  F/ y
ph
21mo2 2312 . 2  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
3 abss 3414 . . . . 5  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  e. 
{ y } ) )
4 elsn 3831 . . . . . . 7  |-  ( x  e.  { y }  <-> 
x  =  y )
54imbi2i 305 . . . . . 6  |-  ( (
ph  ->  x  e.  {
y } )  <->  ( ph  ->  x  =  y ) )
65albii 1576 . . . . 5  |-  ( A. x ( ph  ->  x  e.  { y } )  <->  A. x ( ph  ->  x  =  y ) )
73, 6bitri 242 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  =  y ) )
8 snex 4407 . . . . 5  |-  { y }  e.  _V
98ssex 4349 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  ->  { x  |  ph }  e.  _V )
107, 9sylbir 206 . . 3  |-  ( A. x ( ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
1110exlimiv 1645 . 2  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
122, 11sylbi 189 1  |-  ( E* x ph  ->  { x  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550   E.wex 1551    e. wcel 1726   E*wmo 2284   {cab 2424   _Vcvv 2958    C_ wss 3322   {csn 3816
This theorem is referenced by:  rmorabex  4425  euabex  4426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-sn 3822  df-pr 3823
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