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Theorem riotasbc 5605
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 3106 . . 3  |-  { x  e.  A  |  ph }  C_ 
{ x  |  ph }
2 riotacl2 5603 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  { x  e.  A  |  ph }
)
31, 2sseldi 3021 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  { x  |  ph } )
4 df-sbc 2839 . 2  |-  ( [. ( iota_ x  e.  A  ph )  /  x ]. ph  <->  (
iota_ x  e.  A  ph )  e.  { x  |  ph } )
53, 4sylibr 132 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 1438   {cab 2074   E!wreu 2361   {crab 2363   [.wsbc 2838   iota_crio 5589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-uni 3649  df-iota 4967  df-riota 5590
This theorem is referenced by:  riotass2  5616  riotass  5617  cjth  10245
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