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Theorem riota1 5716
Description: Property of restricted iota. Compare iota1 5072. (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 2400 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 iota1 5072 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
31, 2sylbi 120 . 2  |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
4 df-riota 5698 . . 3  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
54eqeq1i 2125 . 2  |-  ( (
iota_ x  e.  A  ph )  =  x  <->  ( iota x ( x  e.  A  /\  ph )
)  =  x )
63, 5syl6bbr 197 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    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   E!weu 1977   E!wreu 2395   iotacio 5056   iota_crio 5697
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-reu 2400  df-v 2662  df-sbc 2883  df-un 3045  df-sn 3503  df-pr 3504  df-uni 3707  df-iota 5058  df-riota 5698
This theorem is referenced by:  supelti  6857  oddpwdclemdvds  11775  oddpwdclemndvds  11776
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