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Theorem euiotaex 4983
Description: Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the  iota class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.)
Assertion
Ref Expression
euiotaex  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )

Proof of Theorem euiotaex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iotaval 4978 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
21eqcomd 2093 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
32eximi 1536 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y  y  =  ( iota x ph ) )
4 df-eu 1951 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
5 isset 2625 . 2  |-  ( ( iota x ph )  e.  _V  <->  E. y  y  =  ( iota x ph ) )
63, 4, 53imtr4i 199 1  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   E!weu 1948   _Vcvv 2619   iotacio 4965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649  df-iota 4967
This theorem is referenced by:  iota4an  4986  funfvex  5306
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