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Theorem riotaprop 6344
Description: Properties of a restricted definite description operator. Todo: can some uses of riota2f 6342 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.3  |-  ( 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 6335 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  A
)
31, 2syl5eqel 2380 . 2  |-  ( E! x  e.  A  ph  ->  B  e.  A )
41eqcomi 2300 . . . 4  |-  ( iota_ x  e.  A ph )  =  B
5 nfriota1 6328 . . . . . 6  |-  F/_ x
( iota_ x  e.  A ph )
61, 5nfcxfr 2429 . . . . 5  |-  F/_ x B
7 riotaprop.0 . . . . 5  |-  F/ x ps
8 riotaprop.3 . . . . 5  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
96, 7, 8riota2f 6342 . . . 4  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  (
iota_ x  e.  A ph )  =  B
) )
104, 9mpbiri 224 . . 3  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ps )
113, 10mpancom 650 . 2  |-  ( E! x  e.  A  ph  ->  ps )
123, 11jca 518 1  |-  ( E! x  e.  A  ph  ->  ( B  e.  A  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1534    = wceq 1632    e. wcel 1696   E!wreu 2558   iota_crio 6313
This theorem is referenced by:  fin23lem27  7970  lble  9722  lineval22  26185  ltrniotaval  31392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-riota 6320
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