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Theorem iotaexeu 36839
Description: The iota class exists. This theorem does not require ax-nul 4527 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaexeu  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )

Proof of Theorem iotaexeu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iotaval 5564 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
21eqcomd 2477 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
32eximi 1715 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y  y  =  ( iota x ph ) )
4 df-eu 2323 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
5 isset 3035 . 2  |-  ( ( iota x ph )  e.  _V  <->  E. y  y  =  ( iota x ph ) )
63, 4, 53imtr4i 274 1  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904   E!weu 2319   _Vcvv 3031   iotacio 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-v 3033  df-sbc 3256  df-un 3395  df-sn 3960  df-pr 3962  df-uni 4191  df-iota 5553
This theorem is referenced by:  iotasbc  36840  pm14.18  36849  iotavalb  36851  sbiota1  36855
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