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Theorem riotaprop 6188
Description: Properties of a restricted definite description operator. Todo (df-riota 6164 update): can some uses of riota2f 6186 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 6179 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  A )
31, 2syl5eqel 2546 . 2  |-  ( E! x  e.  A  ph  ->  B  e.  A )
41eqcomi 2467 . . . 4  |-  ( iota_ x  e.  A  ph )  =  B
5 nfriota1 6171 . . . . . 6  |-  F/_ x
( iota_ x  e.  A  ph )
61, 5nfcxfr 2614 . . . . 5  |-  F/_ x B
7 riotaprop.0 . . . . 5  |-  F/ x ps
8 riotaprop.2 . . . . 5  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
96, 7, 8riota2f 6186 . . . 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 669 . 2  |-  ( E! x  e.  A  ph  ->  ps )
123, 11jca 532 1  |-  ( E! x  e.  A  ph  ->  ( B  e.  A  /\  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   F/wnf 1590    e. wcel 1758   E!wreu 2801   iota_crio 6163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-un 3444  df-in 3446  df-ss 3453  df-sn 3989  df-pr 3991  df-uni 4203  df-iota 5492  df-riota 6164
This theorem is referenced by:  fin23lem27  8611  lble  10396  ltrniotaval  34583
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