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Theorem riota1a 6536
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 491 . 2  |-  ( x  e.  A  ->  ( ph 
<->  ( x  e.  A  /\  ph ) ) )
2 df-reu 2681 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 iota1 5399 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
42, 3sylbi 188 . 2  |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
51, 4sylan9bb 681 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   E!weu 2262   E!wreu 2676   iotacio 5383
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
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 2266  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-reu 2681  df-v 2926  df-sbc 3130  df-un 3293  df-sn 3788  df-pr 3789  df-uni 3984  df-iota 5385
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