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Theorem riotasbc 5859
Description: Substitution law for descriptions. (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 3255 . . 3  |-  { x  e.  A  |  ph }  C_ 
{ x  |  ph }
2 riotacl2 5857 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  { x  e.  A  |  ph }
)
31, 2sselid 3165 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  { x  |  ph } )
4 df-sbc 2975 . 2  |-  ( [. ( iota_ x  e.  A  ph )  /  x ]. ph  <->  (
iota_ x  e.  A  ph )  e.  { x  |  ph } )
53, 4sylibr 134 1  |-  ( E! x  e.  A  ph  ->  [. ( iota_ x  e.  A  ph )  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2158   {cab 2173   E!wreu 2467   {crab 2469   [.wsbc 2974   iota_crio 5843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-uni 3822  df-iota 5190  df-riota 5844
This theorem is referenced by:  riotass2  5870  riotass  5871  cjth  10868
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