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Theorem euiotaex 4907
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 4902 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
21eqcomd 2087 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
32eximi 1532 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y  y  =  ( iota x ph ) )
4 df-eu 1945 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
5 isset 2606 . 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 1283    = wceq 1285   E.wex 1422    e. wcel 1434   E!weu 1942   _Vcvv 2602   iotacio 4889
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-sn 3406  df-pr 3407  df-uni 3604  df-iota 4891
This theorem is referenced by:  iota4an  4910  funfvex  5217
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