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Theorem iota4 5582
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
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-eu 2313 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 biimpr 203 . . . . . 6  |-  ( (
ph 
<->  x  =  z )  ->  ( x  =  z  ->  ph ) )
32alimi 1694 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  A. x
( x  =  z  ->  ph ) )
4 sb2 2193 . . . . 5  |-  ( A. x ( x  =  z  ->  ph )  ->  [ z  /  x ] ph )
53, 4syl 17 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  [ z  /  x ] ph )
6 iotaval 5575 . . . . . 6  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
76eqcomd 2467 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  z  =  ( iota x ph ) )
8 dfsbcq2 3281 . . . . 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 215 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  [. ( iota x ph )  /  x ]. ph )
1110exlimiv 1786 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  [. ( iota x ph )  /  x ]. ph )
121, 11sylbi 200 1  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1452    = wceq 1454   E.wex 1673   [wsb 1807   E!weu 2309   [.wsbc 3278   iotacio 5562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-rex 2754  df-v 3058  df-sbc 3279  df-un 3420  df-sn 3980  df-pr 3982  df-uni 4212  df-iota 5564
This theorem is referenced by:  iota4an  5583  iotacl  5587  pm14.24  36826  sbiota1  36828
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