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Theorem riotasbc 5862
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 3258 . . 3  |-  { x  e.  A  |  ph }  C_ 
{ x  |  ph }
2 riotacl2 5860 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  { x  e.  A  |  ph }
)
31, 2sselid 3168 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  { x  |  ph } )
4 df-sbc 2978 . 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 2160   {cab 2175   E!wreu 2470   {crab 2472   [.wsbc 2977   iota_crio 5846
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-uni 3825  df-iota 5193  df-riota 5847
This theorem is referenced by:  riotass2  5873  riotass  5874  cjth  10873
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