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Theorem iota1 5547
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 2242 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 sp 1883 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  x  =  z ) )
3 iotaval 5544 . . . . . 6  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
43eqeq2d 2416 . . . . 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 2411 . . . 4  |-  ( x  =  ( iota x ph )  <->  ( iota x ph )  =  x
)
75, 6syl6bb 261 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  ( iota x ph )  =  x )
)
87exlimiv 1743 . 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 1403    = wceq 1405   E.wex 1633   E!weu 2238   iotacio 5531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2760  df-v 3061  df-sbc 3278  df-un 3419  df-sn 3973  df-pr 3975  df-uni 4192  df-iota 5533
This theorem is referenced by:  iota2df  5557  sniota  5560  tz6.12-1  5865  opabiota  5912  riota1  6258  riota1a  6259  erovlem  7444  gsumval3OLD  17232  gsumval3lem2  17234  bnj1366  29215
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