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Theorem riotaprop 6573
Description: Properties of a restricted definite description operator. Todo: can some uses of riota2f 6571 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 6564 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  A
)
31, 2syl5eqel 2520 . 2  |-  ( E! x  e.  A  ph  ->  B  e.  A )
41eqcomi 2440 . . . 4  |-  ( iota_ x  e.  A ph )  =  B
5 nfriota1 6557 . . . . . 6  |-  F/_ x
( iota_ x  e.  A ph )
61, 5nfcxfr 2569 . . . . 5  |-  F/_ x B
7 riotaprop.0 . . . . 5  |-  F/ x ps
8 riotaprop.3 . . . . 5  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
96, 7, 8riota2f 6571 . . . 4  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  (
iota_ x  e.  A ph )  =  B
) )
104, 9mpbiri 225 . . 3  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ps )
113, 10mpancom 651 . 2  |-  ( E! x  e.  A  ph  ->  ps )
123, 11jca 519 1  |-  ( E! x  e.  A  ph  ->  ( B  e.  A  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   E!wreu 2707   iota_crio 6542
This theorem is referenced by:  fin23lem27  8208  lble  9960  ltrniotaval  31378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-riota 6549
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