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Theorem riota1a 6258
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 504 . 2  |-  ( x  e.  A  ->  ( ph 
<->  ( x  e.  A  /\  ph ) ) )
2 df-reu 2798 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 iota1 5551 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
42, 3sylbi 195 . 2  |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
51, 4sylan9bb 699 1  |-  ( ( x  e.  A  /\  E! x  e.  A  ph )  ->  ( ph  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   E!weu 2266   E!wreu 2793   iotacio 5535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-rex 2797  df-reu 2798  df-v 3095  df-sbc 3312  df-un 3463  df-sn 4011  df-pr 4013  df-uni 4231  df-iota 5537
This theorem is referenced by: (None)
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