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Theorem riota1 5842
Description: Property of restricted iota. Compare iota1 5187. (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 2462 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 iota1 5187 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
31, 2sylbi 121 . 2  |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
4 df-riota 5824 . . 3  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
54eqeq1i 2185 . 2  |-  ( (
iota_ x  e.  A  ph )  =  x  <->  ( iota x ( x  e.  A  /\  ph )
)  =  x )
63, 5bitr4di 198 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 104    <-> wb 105    = wceq 1353   E!weu 2026    e. wcel 2148   E!wreu 2457   iotacio 5171   iota_crio 5823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-reu 2462  df-v 2739  df-sbc 2963  df-un 3133  df-sn 3597  df-pr 3598  df-uni 3808  df-iota 5173  df-riota 5824
This theorem is referenced by:  supelti  6994  oddpwdclemdvds  12140  oddpwdclemndvds  12141
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