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Theorem iota1 5410
Description: Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
iota1  |-  ( E! x ph  ->  ( ph 
<->  ( iota x ph )  =  x )
)

Proof of Theorem iota1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-eu 2257 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 sp 1794 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  x  =  z ) )
3 iotaval 5407 . . . . . 6  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
43eqeq2d 2454 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  (
x  =  ( iota
x ph )  <->  x  =  z ) )
52, 4bitr4d 256 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  x  =  ( iota
x ph ) ) )
6 eqcom 2445 . . . 4  |-  ( x  =  ( iota x ph )  <->  ( iota x ph )  =  x
)
75, 6syl6bb 261 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  ( iota x ph )  =  x )
)
87exlimiv 1688 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  ( ph  <->  ( iota x ph )  =  x ) )
91, 8sylbi 195 1  |-  ( E! x ph  ->  ( ph 
<->  ( iota x ph )  =  x )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1367    = wceq 1369   E.wex 1586   E!weu 2253   iotacio 5394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rex 2736  df-v 2989  df-sbc 3202  df-un 3348  df-sn 3893  df-pr 3895  df-uni 4107  df-iota 5396
This theorem is referenced by:  iota2df  5420  sniota  5423  tz6.12-1  5721  opabiota  5769  riota1  6086  riota1a  6087  erovlem  7211  gsumval3OLD  16397  gsumval3lem2  16399  bnj1366  31842
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