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Theorem riota1 6560
Description: Property of restricted iota. Compare iota1 5424. (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 2704 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 iota1 5424 . . 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 6547 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
54eqeq1d 2443 . 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 1652    e. wcel 1725   E!weu 2280   E!wreu 2699   iotacio 5408   iota_crio 6534
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 2416
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 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-reu 2704  df-v 2950  df-sbc 3154  df-un 3317  df-if 3732  df-sn 3812  df-pr 3813  df-uni 4008  df-iota 5410  df-riota 6541
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