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Theorem iotaex 5566
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 5560 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
21eqcomd 2475 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  z  =  ( iota x ph ) )
32eximi 1635 . . 3  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  E. z  z  =  ( iota x ph ) )
4 df-eu 2279 . . 3  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
5 isset 3117 . . 3  |-  ( ( iota x ph )  e.  _V  <->  E. z  z  =  ( iota x ph ) )
63, 4, 53imtr4i 266 . 2  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
7 iotanul 5564 . . 3  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
8 0ex 4577 . . 3  |-  (/)  e.  _V
97, 8syl6eqel 2563 . 2  |-  ( -.  E! x ph  ->  ( iota x ph )  e.  _V )
106, 9pm2.61i 164 1  |-  ( iota
x ph )  e.  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275   _Vcvv 3113   (/)c0 3785   iotacio 5547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-pr 4030  df-uni 4246  df-iota 5549
This theorem is referenced by:  iota4an  5568  fvex  5874  riotaex  6247  erov  7405  iunfictbso  8491  isf32lem9  8737  sumex  13469  pcval  14223  grpidval  15745  fn0g  15746  gsumvalx  15815  psgnfn  16322  psgnval  16328  dchrptlem1  23267  lgsdchrval  23350  lgsdchr  23351  prodex  28616  ellimciota  31156  fourierdlem36  31443  fourierdlem54  31461  bnj1366  32967  bj-finsumval0  33735
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