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Theorem moabex 4422
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 1629 . . 3  |-  F/ y
ph
21mo2 2310 . 2  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
3 abss 3412 . . . . 5  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  e. 
{ y } ) )
4 elsn 3829 . . . . . . 7  |-  ( x  e.  { y }  <-> 
x  =  y )
54imbi2i 304 . . . . . 6  |-  ( (
ph  ->  x  e.  {
y } )  <->  ( ph  ->  x  =  y ) )
65albii 1575 . . . . 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 4405 . . . . 5  |-  { y }  e.  _V
98ssex 4347 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  ->  { x  |  ph }  e.  _V )
107, 9sylbir 205 . . 3  |-  ( A. x ( ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
1110exlimiv 1644 . 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 1549   E.wex 1550    e. wcel 1725   E*wmo 2282   {cab 2422   _Vcvv 2956    C_ wss 3320   {csn 3814
This theorem is referenced by:  rmorabex  4423  euabex  4424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-sn 3820  df-pr 3821
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