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Theorem iota1 5233
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 2147 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 sp 1716 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  x  =  z ) )
3 iotaval 5230 . . . . . 6  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
43eqeq2d 2294 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  (
x  =  ( iota
x ph )  <->  x  =  z ) )
52, 4bitr4d 247 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  x  =  ( iota
x ph ) ) )
6 eqcom 2285 . . . 4  |-  ( x  =  ( iota x ph )  <->  ( iota x ph )  =  x
)
75, 6syl6bb 252 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  ( iota x ph )  =  x )
)
87exlimiv 1666 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  ( ph  <->  ( iota x ph )  =  x ) )
91, 8sylbi 187 1  |-  ( E! x ph  ->  ( ph 
<->  ( iota x ph )  =  x )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528    = wceq 1623   E!weu 2143   iotacio 5217
This theorem is referenced by:  iota2df  5243  sniota  5246  tz6.12-1  5544  opabiota  6293  riota1  6323  riota1a  6324  erovlem  6754  gsumval3  15191  bnj1366  28862
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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