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Theorem iota4 5403
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 2266 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 bi2 190 . . . . . 6  |-  ( (
ph 
<->  x  =  z )  ->  ( x  =  z  ->  ph ) )
32alimi 1565 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  A. x
( x  =  z  ->  ph ) )
4 sb2 2079 . . . . 5  |-  ( A. x ( x  =  z  ->  ph )  ->  [ z  /  x ] ph )
53, 4syl 16 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  [ z  /  x ] ph )
6 iotaval 5396 . . . . . 6  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
76eqcomd 2417 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  z  =  ( iota x ph ) )
8 dfsbcq2 3132 . . . . 5  |-  ( z  =  ( iota x ph )  ->  ( [ z  /  x ] ph 
<-> 
[. ( iota x ph )  /  x ]. ph ) )
97, 8syl 16 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  ( [ z  /  x ] ph  <->  [. ( iota x ph )  /  x ]. ph ) )
105, 9mpbid 202 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  [. ( iota x ph )  /  x ]. ph )
1110exlimiv 1641 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  [. ( iota x ph )  /  x ]. ph )
121, 11sylbi 188 1  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546   E.wex 1547    = wceq 1649   [wsb 1655   E!weu 2262   [.wsbc 3129   iotacio 5383
This theorem is referenced by:  iota4an  5404  iotacl  5408  pm14.24  27508  sbiota1  27510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-v 2926  df-sbc 3130  df-un 3293  df-sn 3788  df-pr 3789  df-uni 3984  df-iota 5385
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