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Theorem iota4 5581
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 2270 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 biimpr 202 . . . . . 6  |-  ( (
ph 
<->  x  =  z )  ->  ( x  =  z  ->  ph ) )
32alimi 1681 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  A. x
( x  =  z  ->  ph ) )
4 sb2 2147 . . . . 5  |-  ( A. x ( x  =  z  ->  ph )  ->  [ z  /  x ] ph )
53, 4syl 17 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  [ z  /  x ] ph )
6 iotaval 5574 . . . . . 6  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
76eqcomd 2431 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  z  =  ( iota x ph ) )
8 dfsbcq2 3303 . . . . 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 214 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  [. ( iota x ph )  /  x ]. ph )
1110exlimiv 1767 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  [. ( iota x ph )  /  x ]. ph )
121, 11sylbi 199 1  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1436    = wceq 1438   E.wex 1660   [wsb 1787   E!weu 2266   [.wsbc 3300   iotacio 5561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-rex 2782  df-v 3084  df-sbc 3301  df-un 3442  df-sn 3998  df-pr 4000  df-uni 4218  df-iota 5563
This theorem is referenced by:  iota4an  5582  iotacl  5586  pm14.24  36685  sbiota1  36687
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