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Theorem riotaprop 6262
Description: Properties of a restricted definite description operator. Todo (df-riota 6239 update): can some uses of riota2f 6260 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 6253 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  A )
31, 2syl5eqel 2494 . 2  |-  ( E! x  e.  A  ph  ->  B  e.  A )
41eqcomi 2415 . . . 4  |-  ( iota_ x  e.  A  ph )  =  B
5 nfriota1 6246 . . . . . 6  |-  F/_ x
( iota_ x  e.  A  ph )
61, 5nfcxfr 2562 . . . . 5  |-  F/_ x B
7 riotaprop.0 . . . . 5  |-  F/ x ps
8 riotaprop.2 . . . . 5  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
96, 7, 8riota2f 6260 . . . 4  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  (
iota_ x  e.  A  ph )  =  B ) )
104, 9mpbiri 233 . . 3  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ps )
113, 10mpancom 667 . 2  |-  ( E! x  e.  A  ph  ->  ps )
123, 11jca 530 1  |-  ( E! x  e.  A  ph  ->  ( B  e.  A  /\  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   F/wnf 1637    e. wcel 1842   E!wreu 2755   iota_crio 6238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-un 3418  df-in 3420  df-ss 3427  df-sn 3972  df-pr 3974  df-uni 4191  df-iota 5532  df-riota 6239
This theorem is referenced by:  fin23lem27  8739  lble  10534  ltrniotaval  33580
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