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Theorem riota1a 6572
Description: Property of iota. (Contributed by NM, 23-Aug-2011.)
Assertion
Ref Expression
riota1a  |-  ( ( x  e.  A  /\  E! x  e.  A  ph )  ->  ( ph  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )

Proof of Theorem riota1a
StepHypRef Expression
1 ibar 492 . 2  |-  ( x  e.  A  ->  ( ph 
<->  ( x  e.  A  /\  ph ) ) )
2 df-reu 2714 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 iota1 5435 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
42, 3sylbi 189 . 2  |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
51, 4sylan9bb 682 1  |-  ( ( x  e.  A  /\  E! x  e.  A  ph )  ->  ( ph  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E!weu 2283   E!wreu 2709   iotacio 5419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-reu 2714  df-v 2960  df-sbc 3164  df-un 3327  df-sn 3822  df-pr 3823  df-uni 4018  df-iota 5421
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