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Theorem iotauni 5433
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 2287 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 iotaval 5432 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
3 uniabio 5431 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  U. {
x  |  ph }  =  z )
42, 3eqtr4d 2473 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  = 
U. { x  | 
ph } )
54exlimiv 1645 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  ( iota x ph )  =  U. { x  |  ph }
)
61, 5sylbi 189 1  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550   E.wex 1551    = wceq 1653   E!weu 2283   {cab 2424   U.cuni 4017   iotacio 5419
This theorem is referenced by:  iotaint  5434  iotassuni  5437  dfiota4  5449  fveu  5723  riotauni  6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-v 2960  df-sbc 3164  df-un 3327  df-sn 3822  df-pr 3823  df-uni 4018  df-iota 5421
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