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Theorem iota4 5237
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 2147 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 bi2 189 . . . . . 6  |-  ( (
ph 
<->  x  =  z )  ->  ( x  =  z  ->  ph ) )
32alimi 1546 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  A. x
( x  =  z  ->  ph ) )
4 sb2 1963 . . . . 5  |-  ( A. x ( x  =  z  ->  ph )  ->  [ z  /  x ] ph )
53, 4syl 15 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  [ z  /  x ] ph )
6 iotaval 5230 . . . . . 6  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
76eqcomd 2288 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  z  =  ( iota x ph ) )
8 dfsbcq2 2994 . . . . 5  |-  ( z  =  ( iota x ph )  ->  ( [ z  /  x ] ph 
<-> 
[. ( iota x ph )  /  x ]. ph ) )
97, 8syl 15 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  ( [ z  /  x ] ph  <->  [. ( iota x ph )  /  x ]. ph ) )
105, 9mpbid 201 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  [. ( iota x ph )  /  x ]. ph )
1110exlimiv 1666 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  [. ( iota x ph )  /  x ]. ph )
121, 11sylbi 187 1  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528    = wceq 1623   [wsb 1629   E!weu 2143   [.wsbc 2991   iotacio 5217
This theorem is referenced by:  iota4an  5238  iotacl  5242  pm14.24  27632  sbiota1  27634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-sbc 2992  df-un 3157  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219
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