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Theorem riotasbc 6280
Description: Substitution law for descriptions. Compare iotasbc 36672. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotasbc  |-  ( E! x  e.  A  ph  ->  [. ( iota_ x  e.  A  ph )  /  x ]. ph )

Proof of Theorem riotasbc
StepHypRef Expression
1 rabssab 3549 . . 3  |-  { x  e.  A  |  ph }  C_ 
{ x  |  ph }
2 riotacl2 6278 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  { x  e.  A  |  ph }
)
31, 2sseldi 3463 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  { x  |  ph } )
4 df-sbc 3301 . 2  |-  ( [. ( iota_ x  e.  A  ph )  /  x ]. ph  <->  (
iota_ x  e.  A  ph )  e.  { x  |  ph } )
53, 4sylibr 216 1  |-  ( E! x  e.  A  ph  ->  [. ( iota_ x  e.  A  ph )  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1869   {cab 2408   E!wreu 2778   {crab 2780   [.wsbc 3300   iota_crio 6264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-un 3442  df-in 3444  df-ss 3451  df-sn 3998  df-pr 4000  df-uni 4218  df-iota 5563  df-riota 6265
This theorem is referenced by:  riotass2  6291  riotass  6292  cjth  13160  joinlem  16250  meetlem  16264  finxpreclem4  31744  poimirlem26  31924  riotasvd  32491  lshpkrlem3  32641
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