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Theorem iota4 6208
Description: Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iota4  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )

Proof of Theorem iota4
StepHypRef Expression
1 df-eu 2121 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 bi2 191 . . . . . 6  |-  ( (
ph 
<->  x  =  z )  ->  ( x  =  z  ->  ph ) )
32alimi 1546 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  A. x
( x  =  z  ->  ph ) )
4 sb2 1889 . . . . 5  |-  ( A. x ( x  =  z  ->  ph )  ->  [ z  /  x ] ph )
53, 4syl 17 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  [ z  /  x ] ph )
6 iotaval 6201 . . . . . 6  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
76eqcomd 2261 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  z  =  ( iota x ph ) )
8 dfsbcq2 2938 . . . . 5  |-  ( z  =  ( iota x ph )  ->  ( [ z  /  x ] ph 
<-> 
[. ( iota x ph )  /  x ]. ph ) )
97, 8syl 17 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  ( [ z  /  x ] ph  <->  [. ( iota x ph )  /  x ]. ph ) )
105, 9mpbid 203 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  [. ( iota x ph )  /  x ]. ph )
1110exlimiv 2024 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  [. ( iota x ph )  /  x ]. ph )
121, 11sylbi 189 1  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532   E.wex 1537    = wceq 1619   [wsb 1883   E!weu 2117   [.wsbc 2935   iotacio 6188
This theorem is referenced by:  iota4an  6209  iotacl  6213  pm14.24  26965  sbiota1  26967
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-rex 2521  df-v 2742  df-sbc 2936  df-un 3099  df-sn 3587  df-pr 3588  df-uni 3769  df-iota 6190
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