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Theorem riota1 6323
Description: Property of restricted iota. Compare iota1 5233. (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 2550 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 iota1 5233 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
31, 2sylbi 187 . 2  |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
4 riotaiota 6310 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
54eqeq1d 2291 . 2  |-  ( E! x  e.  A  ph  ->  ( ( iota_ x  e.  A ph )  =  x  <->  ( iota x
( x  e.  A  /\  ph ) )  =  x ) )
63, 5bitr4d 247 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E!weu 2143   E!wreu 2545   iotacio 5217   iota_crio 6297
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-reu 2550  df-v 2790  df-sbc 2992  df-un 3157  df-if 3566  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219  df-riota 6304
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