MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iota1 Structured version   Unicode version

Theorem iota1 5432
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 2285 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 sp 1763 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  x  =  z ) )
3 iotaval 5429 . . . . . 6  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
43eqeq2d 2447 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  (
x  =  ( iota
x ph )  <->  x  =  z ) )
52, 4bitr4d 248 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  x  =  ( iota
x ph ) ) )
6 eqcom 2438 . . . 4  |-  ( x  =  ( iota x ph )  <->  ( iota x ph )  =  x
)
75, 6syl6bb 253 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  ( ph 
<->  ( iota x ph )  =  x )
)
87exlimiv 1644 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  ( ph  <->  ( iota x ph )  =  x ) )
91, 8sylbi 188 1  |-  ( E! x ph  ->  ( ph 
<->  ( iota x ph )  =  x )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550    = wceq 1652   E!weu 2281   iotacio 5416
This theorem is referenced by:  iota2df  5442  sniota  5445  tz6.12-1  5747  opabiota  6538  riota1  6568  riota1a  6569  erovlem  7000  gsumval3  15514  bnj1366  29201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-v 2958  df-sbc 3162  df-un 3325  df-sn 3820  df-pr 3821  df-uni 4016  df-iota 5418
  Copyright terms: Public domain W3C validator