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Theorem iotauni 4930
Description: Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iotauni  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )

Proof of Theorem iotauni
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-eu 1946 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 iotaval 4929 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
3 uniabio 4928 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  U. {
x  |  ph }  =  z )
42, 3eqtr4d 2118 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  = 
U. { x  | 
ph } )
54exlimiv 1530 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  ( iota x ph )  =  U. { x  |  ph }
)
61, 5sylbi 119 1  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422   E!weu 1943   {cab 2069   U.cuni 3622   iotacio 4916
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2613  df-sbc 2826  df-un 2987  df-sn 3423  df-pr 3424  df-uni 3623  df-iota 4918
This theorem is referenced by:  iotaint  4931  fveu  5223  riotauni  5527
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