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Theorem riotaprop 5542
Description: Properties of a restricted definite description operator. Todo (df-riota 5519 update): can some uses of riota2f 5540 be shortened with this? (Contributed by NM, 23-Nov-2013.)
Hypotheses
Ref Expression
riotaprop.0  |-  F/ x ps
riotaprop.1  |-  B  =  ( iota_ x  e.  A  ph )
riotaprop.2  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
riotaprop  |-  ( E! x  e.  A  ph  ->  ( B  e.  A  /\  ps ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    B( x)

Proof of Theorem riotaprop
StepHypRef Expression
1 riotaprop.1 . . 3  |-  B  =  ( iota_ x  e.  A  ph )
2 riotacl 5533 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  A )
31, 2syl5eqel 2169 . 2  |-  ( E! x  e.  A  ph  ->  B  e.  A )
41eqcomi 2087 . . . 4  |-  ( iota_ x  e.  A  ph )  =  B
5 nfriota1 5526 . . . . . 6  |-  F/_ x
( iota_ x  e.  A  ph )
61, 5nfcxfr 2220 . . . . 5  |-  F/_ x B
7 riotaprop.0 . . . . 5  |-  F/ x ps
8 riotaprop.2 . . . . 5  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
96, 7, 8riota2f 5540 . . . 4  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  (
iota_ x  e.  A  ph )  =  B ) )
104, 9mpbiri 166 . . 3  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ps )
113, 10mpancom 413 . 2  |-  ( E! x  e.  A  ph  ->  ps )
123, 11jca 300 1  |-  ( E! x  e.  A  ph  ->  ( B  e.  A  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   F/wnf 1390    e. wcel 1434   E!wreu 2355   iota_crio 5518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-uni 3622  df-iota 4917  df-riota 5519
This theorem is referenced by:  lble  8144
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