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Theorem iotaexeu 27586
Description: The iota class exists. This theorem does not require ax-nul 4330 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 5421 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
21eqcomd 2440 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
32eximi 1585 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y  y  =  ( iota x ph ) )
4 df-eu 2284 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
5 isset 2952 . 2  |-  ( ( iota x ph )  e.  _V  <->  E. y  y  =  ( iota x ph ) )
63, 4, 53imtr4i 258 1  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2280   _Vcvv 2948   iotacio 5408
This theorem is referenced by:  iotasbc  27587  pm14.18  27596  iotavalb  27598  sbiota1  27602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-v 2950  df-sbc 3154  df-un 3317  df-sn 3812  df-pr 3813  df-uni 4008  df-iota 5410
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