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Theorem iotaval 4821
Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaval  iota
Distinct variable group:   ,
Allowed substitution hints:   (,)

Proof of Theorem iotaval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfiota2 4811 . 2  iota  U. {  |  }
2 vex 2554 . . . . . . 7 
_V
3 sbeqalb 2809 . . . . . . . 8  _V
4 equcomi 1589 . . . . . . . 8
53, 4syl6 29 . . . . . . 7  _V
62, 5ax-mp 7 . . . . . 6
76ex 108 . . . . 5
8 equequ2 1596 . . . . . . . . . 10
98equcoms 1591 . . . . . . . . 9
109bibi2d 221 . . . . . . . 8
1110biimpd 132 . . . . . . 7
1211alimdv 1756 . . . . . 6
1312com12 27 . . . . 5
147, 13impbid 120 . . . 4
1514alrimiv 1751 . . 3
16 uniabio 4820 . . 3  U. {  |  }
1715, 16syl 14 . 2  U. {  |  }
181, 17syl5eq 2081 1  iota
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   wceq 1242   wcel 1390   {cab 2023   _Vcvv 2551   U.cuni 3571   iotacio 4808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810
This theorem is referenced by:  iotauni  4822  iota1  4824  euiotaex  4826  iota4  4828  iota5  4830
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