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Theorem iota1 5193
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 2029 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 sp 1511 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  x  =  z ) )
3 iotaval 5190 . . . . . 6  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
43eqeq2d 2189 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  (
x  =  ( iota
x ph )  <->  x  =  z ) )
52, 4bitr4d 191 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  x  =  ( iota
x ph ) ) )
6 eqcom 2179 . . . 4  |-  ( x  =  ( iota x ph )  <->  ( iota x ph )  =  x
)
75, 6bitrdi 196 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  ( iota x ph )  =  x )
)
87exlimiv 1598 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  ( ph  <->  ( iota x ph )  =  x ) )
91, 8sylbi 121 1  |-  ( E! x ph  ->  ( ph 
<->  ( iota x ph )  =  x )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351    = wceq 1353   E.wex 1492   E!weu 2026   iotacio 5177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-sn 3599  df-pr 3600  df-uni 3811  df-iota 5179
This theorem is referenced by:  iota2df  5203  sniota  5208  tz6.12-1  5543  riota1  5849  riota1a  5850  erovlem  6627
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