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Theorem riota1a 5752
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 299 . 2  |-  ( x  e.  A  ->  ( ph 
<->  ( x  e.  A  /\  ph ) ) )
2 df-reu 2423 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 iota1 5105 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
42, 3sylbi 120 . 2  |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
51, 4sylan9bb 457 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    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   E!weu 1999   E!wreu 2418   iotacio 5089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-reu 2423  df-v 2688  df-sbc 2910  df-un 3075  df-sn 3533  df-pr 3534  df-uni 3740  df-iota 5091
This theorem is referenced by: (None)
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