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Theorem iotaex 6270
Description: Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the  iota class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaex  |-  ( iota
x ph )  e.  _V

Proof of Theorem iotaex
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 iotaval 6264 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
21eqcomd 2289 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  z  =  ( iota x ph ) )
32eximi 1563 . . 3  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  E. z  z  =  ( iota x ph ) )
4 df-eu 2148 . . 3  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
5 isset 2793 . . 3  |-  ( ( iota x ph )  e.  _V  <->  E. z  z  =  ( iota x ph ) )
63, 4, 53imtr4i 257 . 2  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
7 iotanul 6268 . . 3  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
8 0ex 4151 . . 3  |-  (/)  e.  _V
97, 8syl6eqel 2372 . 2  |-  ( -.  E! x ph  ->  ( iota x ph )  e.  _V )
106, 9pm2.61i 156 1  |-  ( iota
x ph )  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1685   E!weu 2144   _Vcvv 2789   (/)c0 3456   iotacio 6251
This theorem is referenced by:  iota4an  6272  riotaex  6304  erov  6751  iunfictbso  7737  isf32lem9  7983  sumex  12156  pcval  12893  grpidval  14380  fn0g  14381  gsumvalx  14447  dchrptlem1  20499  lgsdchrval  20582  lgsdchr  20583  psgnfn  26835  psgnval  26841  bnj1366  28141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-nul 4150
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-sn 3647  df-pr 3648  df-uni 3829  df-iota 6253
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