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Theorem riota1 6504
Description: Property of restricted iota. Compare iota1 5372. (Contributed by Mario Carneiro, 15-Oct-2016.)
Assertion
Ref Expression
riota1  |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  (
iota_ x  e.  A ph )  =  x
) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem riota1
StepHypRef Expression
1 df-reu 2656 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 iota1 5372 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
31, 2sylbi 188 . 2  |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
4 riotaiota 6491 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
54eqeq1d 2395 . 2  |-  ( E! x  e.  A  ph  ->  ( ( iota_ x  e.  A ph )  =  x  <->  ( iota x
( x  e.  A  /\  ph ) )  =  x ) )
63, 5bitr4d 248 1  |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  (
iota_ x  e.  A ph )  =  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E!weu 2238   E!wreu 2651   iotacio 5356   iota_crio 6478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-reu 2656  df-v 2901  df-sbc 3105  df-un 3268  df-if 3683  df-sn 3763  df-pr 3764  df-uni 3958  df-iota 5358  df-riota 6485
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